INTEILECTUAL     1 


IN  MEMORIAM 
FLORIAN  CAJORI 


INTELLECTUAL 


ARITHMETIC, 


OB, 


flN  ANALYSIS  OF  THE  SCIENCE  OF  NUMBERS. 


WITH    ESPECIAL   REFEHF^CE 


MENTAL  TRAINING  AND  DEVELOPMENT 


BY 

CHARLES  DAVIES,  LL.IX, 

AUTHOR    OF    A   SERIES   OF    ARITHMETICS,  ELEMENTARY    AL<5EBR£, 
ELEMENTS   OF    SURVEYING,    ELEMENTS   OF    DESCTUPTIVS 

GEOMETRY.    SHADES,    SHADOWS,    AND    PERSPECTIVE 

^->  '•'"'« 

ANALYTICAL   GEOMETRY,    AND   DIFFERE 

AND   INTEGRAL   CALCULIIS^,   I 


NEW  YORK: 
PUBLISHED  BY  A.  Sr'B&RNES  &  CO.,  j 

51  &  53  JOHN  STREET. 

BOLD    BT    BOOKSELLKE8,    OP.NRKALLY,    TIIKOUGHOITT   THE   UNITED   STATES. 

1859. 


'T-^DStS 

SUGGESTIONS. 


THIS  work  is  designed  both  for  primary  and  advanced  classes 
The  first  part  is  adapted  to  beginners,  while  the  latter  part  is 
peculiarly  fitted  to  give  to  the  more  advanced  student  that  tko 
rough  mental  drilling,  in  the  Analysis  of  Numbers,  which  fur 
Dishes  the  true  basis  of  all  mathematical  knowledge. 

It  is  suggested  that  classes  in  Higher  Arithmetic,  and  eveu 
in  Algebra,  not  familiar  with  works  of  this  kind,  will  be  greatly 
benefited  by  a  thorough  exercise  in  this  most  important  branch 
of  mathematical  science. 

The.  Teacher  should  require  the  class  to  dispense  with  their 
books  at  the:  time  of  recitation.  He  should  read  each  example, 
and  then  call  upon  some  member  of  the  class  to  solve  it.  The 
Tuvoil  should  rise  and  repeat  the  example  in  the  same  language 
used  'by  the-teacher,  tad  should  then  proceed  to  analyze  it. 

The  analysis  will  be  found  to  consist  of  three  parts ;  two  pro 
positions  and  a  conclusion  ;  thus  : 

What  will  4  barrels  of  cider  cost  at  3  dollars  a  barrel  ? 

IST  PROPOSITION  :  Four  barrels  will  cost  4  times  as  much  as 
1  barrel 

2o  PROPOSITION  :  If  1  barrel  costs  3  dollars,  4  barrels  will  cost 
4  times  3  dollars,  which  are  12  dollars  : 

CONCLUSION  :  Therefore,  4  barrels  of  cider  at  3  dollars  a  bar 
rel,  will  cost  12  dollars. 

The  pupil  should  never  be  allowed  to  omit  either  of  the  steps ; 
and  he  should  be  required  alwavs  to  adhere  strictly  to  a  correct 
and  uniform  phraseology  in  the  analysis. 

The  forms  o£  analysis  are  thought  to  be  of  great  service  belt 
to  the  teacher  and  pupil. 

It  is  also  suggested,  that  the  pupil  be  thoroughly  drilled  in 
Lo?sons  III.  .and  IV.,  Sect  VII,  as  they  afford  very  valuabh 
mental  exercises  and  a  great  variety  of  Arithmetical  processe*. 


CAJORT 

Emtered  according  to  Act  of  Congress,  in  the  year  One  Thousand  EifU 

Hundred  and  Fifty-four, 
BY    CHARLES    DAVIES. 

In  the  Clerk'g  Office  of  the  District  Court  of  the  Southern  District  <if 
New  York. 


INTRODUCTION. 


EVERY  book  of  instruction  should  have  a  specific 
object  to  which  the  entire  work,  both  in  matter  and 
method,  should  strictly  conform. 

It  is  the  object  of  this  book  to  train  and  develop  the 
mind  by  means  of  the  science  of  numbers.  Numbers 
are  the  instruments  here  employed  to  strengthen  the 
memory,  to  cultivate  the  faculty  of  abstraction  and  to 
give  force  and  vigor  to  the  reasoning  powers. 

All  our  ideas  of  numbers, are  either  of  unity  or  of 
multiplicity — unity  being  the  elementary  idea  from 
which  all  others  are  derived. 

A  true  analysis  must  conform  to  the  nature  of 
the  subject  analyzed.  It  must  separate  all  the  ideas 
and  principles  into  their  primary  elements,  and  then 
explain  and  make  manifest  the  laws  by  which  these 
elements  are  connected  with  each  other.  Hence,  the 
analysis  of  numbers  must  begin  with  the  unit  1, — for 
this  is  the  foundation,  and  the  science  is  but  the 
development  of  the  various  processes  by  which  all 
other  numbers  are  derived  from  1,  as  a  base,  and  a 
comparison  of  the  base  1,  with  the  numbers  so  derived. 

Every  number  has  what  we  call  a  base:  that  is, 
"number  being  a  collection  of  things  of  the  same 
kind,"  one  of  these  things  is  the  base  of  the  number ; 
and  this  thing,  is  called  a  unit.  If  we  have  the 


IV  INTRODUCTION. 

ber  3  hundred,  we  may  consider  it  in  several  points  of 
view: 

1  st.  It  is  one  hundred  taken  3  times,  and  if  we  regard 
one  hundred  as  the  base,  then,  the  base  is  taken  3 
times  to  make  up  the  number;  and  100  is  the  unit. 

2nd.  We  may  consider  the  number  as  made  up  cf 
30  tens,  and  if  we  regard  10  as  the  base,  then  the  base 
is  taken  30  times;  and  10  is  the  unit. 

3rd.  We  may  also  consider  the  number  as  made  up 
of  300  ones,  in  which,  the  base  is  1,  and  the  unit  of 
the  number  1. 

Again,  if  we  analyze  the  number, 

cwt.          qr.  Ib.         oz.         dr. 

13  2  20         12          4 

We  see,  that  I  cwt.  is  the  base  of  IScwt. ;  Iqr.  the 
base  of  2gr. ;  lib.  the  base  of  20 Ib. ;  loz.  the  base  of 
1202. ;  and  dr.  the  base  of  4dr..  and  all  these  bases  may 
be  referred  to  1  dram  as  a  primary  base ;  hence,  as  in 
simple  numbers,  every  base  may  be  referred  to  the 
unit  1 :  therefore,  in  every  entire  number,  1  is  the 
primary  base. 

Let  us  see  if  the  same  be  true  in  fractional  numbers. 

If  we  have  the  fraction  |-  it  denotes : 

1st.  That  something  regarded  as  a  whole  has  been 
divided  into  8  equal  parts :  and, 

2nd.  That  7  of  these  parts  are  taken. 

In  this  collection  of  7  things,  (each  of  which  is  -|).  -J- 

is  the  base  of  the  fractional  number  | ;  but  it  is  not 

the  primary  base ;  for  £  implies,  either  |  of  1  or  £  ot 

some  collection  of  1's;  if  a  collection  of  1's  we  call 

tnat  collection  unity,  which  may  be  referred  to  the 

primary  base  1 :  hence,  every  number, either  integrator 

\  fractional,  has  the  unit  1  for  a  primary  base. 

\     A  fractional  number,  therefore,  is  merely  a  collee- 


INTRODUCTION.  V 

tion  of  the  equal  parts  of  unity,  and  to  one  of  these 
parts  we  give  the  name  of  fractional  unit.  The  unit 
which  is  divided  is  called  the  unit  of  the  fraction,  and 
may  be  a  collection  of  units,  (as  what  is  ^  of  40  'I)  or 
it  may  be  the  unit  1. 

The  term  UNITY,  in  mathematical  science,  is  applied 
to  any  number  or  quantity  regarded  as  a  whole:  the 
term  unit,  in  arithmetic,  to  any  number  which  is  used 
as  the  base  of  a  collection.  Thus,  10  is  a  unit  of  the 
second  order,  being  the  base  for  the  collection  of  10's 
100  is  a  unit  of  the  third  order,  being  the  base  for  the 
collection  of  hundreds,  and  similarly  for  other  bases. 
Thus,  also,  in  the  fraction  \,  \  is  the  fractional  unit, 
being  the  fractional  base,  while  the  primary  base  is 
the  unit  1. 

Every  arithmetical  process,  therefore,  has  a  direct 
reference  to  the  unit  1 ;  and  with  this  view  of  the 
subject  before  him,  the  pupil  always  has  the  means  of 
making  a  correct  analysis. 

Addition  is  the  process  of  finding  a  number  which 
shall  contain  as  many  units,  and  no  more,  as  are  found 
in  all  the  numbers  added.  Multiplication  is  taking 
one  number,  called  the  multiplicand,  as  many  times 
as  there  are  units  in  another  number,  called  the 
multiplier,  and  the  number  which  shows  the  result  of 
such  taking,  is  called  the  product :  and  similarly  for 
all  other  arithmetical  processes.! 

A  clear  conception  of  elementary  principles,  by 
wiiich  we  mean,  those  principles  that  result  from  a 
final  analysis,  lies  at  the  foundation  of  all  knowledge, 
It  is  not  till  we  get  such  conceptions,  and  have  learned 
the  laws  by  which  they  are  connected,  that  we  have 
acquired  any  thing  deserving  the  name  of  science. 
1* 


1 


TO  INTRODUCTION. 


To  learn  one  thing  at  a  time — to  leain  that  thing 
thoroughly — and  to  learn  its  connections  with  other 
things  are  the  golden  steps  that  lead  to  the  temple  of 
knowledge. 

It  will  be  seen,  in  Lessons  XVI.  and  XVIT.,  Section 
VJL,  that  UNITY  has  been  employed  to  denote  any 
number  entering  into  an  arithmetical  question.  This 
use  of  unity  affords  a  powerful  means  of  solving  most 
questions  which  otherwise  present  great  difficulties; 
and  is,  it  is  believed,  a  link  of  closer  connection  between 
the  subjects  of  arithmetic  and  algebra,  than  has  before 
been  used. 

It  has  been  the  author's  aim,  in  the  present  work,  to 
treat  the  subject  of  number  in  accordance  with  these 
principles,  and  to  give  to  the  whole  a  scientific  form, 
and  logical  development.  That  he  might  not  fail  in  so 
difficult  and  delicate  an  undertaking,  he  has  defined  all 
the  terms,  and  given  a  full  analysis  of  every  process 
employed. 

The  work  is  complete  in  itself.  It  is  a  mental  ana 
lysis  of  the  science  of  numbers,  designed  to  be 
accessible  to  the  youngest  pupils  because  of  its  simple 
gradations,  and  useful  to  the  advanced  pupil  because 
of  its  scientific  arrangement,  its  logical  connections  and 
its  higher  analysis  of  the  properties  and  relations  of 
u  umbers. 

In  the  preparation  of  this  work,  many  valuable 
suggestions  and  methods  have  been  furnished  by 
practical  teachers.  They  were  cheerfully  offered  and 
thankfully  adopted. 

FISIIKILL  LANDING,  February ^  1854. 


INTELLECTUAL 


SECTION    FIRST. 


LESSON  I. 

Counting. 

One, 
Two, 

Three,.   .    .   

Four, ^    *  *  4t 

Five, 

foix,  .• 

Seven, 

Eight, 

Nine, 

Ten, 

Eleven, 

Twelve, 4^  4t  4t  4*  4t  4t  4t  4t  *'4t  4* 

Thirteen, 
Fourteen, 
Fifteen, 
Sixteen,:    . 
Seventeen, . 
Eighteen,   . 
Nineteen, 
Twenty, 

SUGGESTIONS. — There  is  but  one  simple  idea  in  Arithmetic — it 
is  the  idea  of  the  unit  ONE. 

Any  collection  of  units  is  a  number.  Hence,  every  number  is 
derived  from  one,  and  consequently  has  one  for  a  base.  Count 
ing  is  merely  naming  numbers.  In  this  lesson,  the  names  of 
numbers  are  written  opposite  the  collection. 

How  many  units  in  four?  In  six  apples,  what  is  the  unit! 
"What  is  the  unit  in  seven  pears  ?  How  many  units  in  twelve 
peaches  ?  What  is  the  unit  that  is  counted  in  the  lesson  ?  How 
many  stars  in  the  4th  line  ?  How  many  in  the  14th  ?  ,, 

From  what  ar«  all  numbers  derived  ?  What  is  the  base  of 
every  number  ? 


yf  INTELLECTUAL*  ARITHMETIC.  [fiEC.  I. 

*>   ".    f   I    »  «  •'         w  I    <r  »T>  cj  Q  n  -vr       T  T 


II. 

Figures  from  One  to  Twenty. 
It      ........    ............    * 


7  J 

S* 

*>  7 

10    ' 


is  ^  <. 

19  /^  . 
2  0  jj  e. 

Which  figure  stands  for  two  ?  Which  figure  stands 
for  four'?  Which  figure  stands  for  nine?  Which 
stands  for  eight?  What  stands  for  ten?  What 
stands  for  twelve  ?  What  stands  for  fourteen  1 
What  stands  for  sixteen  ?  What  stands  for  eighteen  ? 
What  stands  for  twenty  ?  What  stands  for  seven 
teen  1  What  stands  for  fifteen  ?  What  stands  for 
nineteen  1  What  stands  for  thirteen? 

SUGGESTIONS.  —  This  lesson  is  intended  to  teach  that  number* 
may  be  expressed  by  figures,  as  well  as  by  words.  The  teacher 
should  explain  to  the  pupil  that  the  figure  2  and  the  word  two, 
have  the  same  meaning,  and  similarly  for  every  fig-  ire  and  ita 
^corresponding  word.  Either  the  figure  or  the  word,  denotes  as 
many  units  us  its  name  points  out 


LES.  HI.]  INTELLECTUAL    ARITHMETIC. 

LESSON    III. 
Figures  from  One  to  One  Hundred. 


Naught     .     .     0    Thirty-four    .  34 

Sixty-eight    .  68 

One     ...     1    Thirty-five     .  35 

Sixty-nine     .  69 

Two     ...     2 

Thirty-six      .  36 

Seventy    .     .  70 

Three  ...     3 

Thirty-seven    37 

Seventy-one  .  71 

Four   ...     4 

Thirty-eight.  38  1  Seventy-two  .  72 

Five    ...     5 

Thirty-nine   .  39  Seventy  -three  73 

Six.     ...     6 

Forty  .     .     .  40  1  Seventy-four    74 

Seven  ...     7 

Forty-one     .  41  Seventy  -five  .  75 

Eight  ...     8 

Forty-two     .  42  Seventy-six  .  76 

Nine..     .     .     9 

Forty-three  .  43  Seventy  -seven  77 

Ten      ...  10 

Forty  -four     .  44  j  Seventy-eight  78 

Eleven     .     .  11 

Forty-five     .  45  Seventy-nine    79 

Twelve     .     .   12 

Forty-six.     .  46 

Eighty      .     .  80 

Thirteen  .     .13 

Forty-seven  .  47 

Eighty  -one    .  81 

Fourteen  .     .14 

Forty-eight  .  48  (Eighty-two   .  82 

Fifteen     .     .15 

Forty-nine    .  49 

Eighty-three    83 

Sixteen     .     .16 

Fifty    ...  50 

Eighty  -four  .  84 

Seventeen     .   17 

Fifty  -one.     .  51 

Eighty-five    .  85 

Eighteen  .     .   18 

Fifty-two      .  52 

Eighty-six     .  86 

Nineteen  .     .  19 

Fifty-three    .  53 

Eighty-seven   81 

Twenty    .     .  20 

Fifty-four      .  54 

Eighty-eight    8S 

Twenty  -one  .  21 

Fifty  -five      .  55 

Eighty-nine  .  89 

Twenty-two  .  22 

Fifty-six  .     .  56 

Ninety     .     .  90 

Twenty-three  23 

Fifty  -seven  .  57 

Ninety-one    .  91 

Twenty-four  .  24 

Fifty-eight    .  58 

Ninety-two   .  92 

Twenty-five  .  25 

Fifty-nine      .  59 

Ninety-three    93 

Twenty-six    .  26 

Sixty   ...  60 

Ninety-four  .  94 

Twenty-seven  27 

Sixty-one      .  61 

Ninety-five    .  95 

Twenty-eight  28 

Sixty-two      .  62 

Ninety-six     .  96 

Twenty-nine    29 

Sixty-three    .  63 

Ninety-seven   97 

Thirty      .     .  30  j  Sixty  -four     .  64 

Ninety-eight    98 

Thirty-one    .  31    Sixty-five      .  65 

Ninety-nine  .  99 

Thirty-two    .  32    Sixty-six  .     .  66 
Thirty-three  .  33   Sixty-seven  .  67 

One  hundred  100 
Two  hundred  200 

SUGGESTION.  —  This  lesson  is  intended  to  teach  that  the  word* 

INTELLECTUAL    ARITHMETIC.  [SEG.  1, 

LESSON    IV. 


10 


II  .   . 

III  .  . 

IV  .  . 

V  .  . 

VI  .  . 

VII  .  . 

VIII  .  . 

IX  .  . 

X  .  . 

XI  .  . 

XII  .  . 
XIII.  . 

XIV  . 

XV  .     . 

XVI  . 

XVII  . 

XVIII  . 

XIX  . 

This  table  is  read,  one  I,  one ;  two  I's,  two  ;  three 
Ps,  three  ;  IV,  four,  &c. 

What  stands  for  two?  What  stands  for  four] 
What  stands  for  five  1  What  stands  for  eight  ? 
What  stands  for  ten  ?  What  stands  for  twenty  ] 
What  stands  for  thirty  ?  What  stands  for  forty  ] 
What  stands  for  fifty  1  What  stands  for  sixty  ? 
What  stands  for  seventy  ?  What  stands  for  eighty  ? 
What  stands  for  ninety  1  What  stands  for  one  hun 
dred  ?  What  stands  for  five  hundred  1  What  for 
one  thousand  ? 

SUGGESTION. — This  lesson  is  intended  to  teach  that  numbers 
may  be  expressed  by  the  Roman  characters,  as  well  as  by  words 
and  by  figures.  Hence,  there  are  three  ways  of  expressing 
.  numbers,  viz. :  by  words,  by  figures,  and  by  letters. 


One 

XX  .     . 

Twenty 

Two 

XXI    '  . 

Twenty-one 

Three 

XXX     . 

Thirty 

Four 

XL    .     . 

Forty 

Five 

L      .     . 

Fifty 

Six 

LX    .     . 

Sixty 

Seven 

LXX     . 

Seventy 

Eight 

LXXX  . 

Eighty 

Nine 

xc  .   . 

Ninety 

Ten 

c     .   . 

One  hundred 

Eleven 

cc  .   . 

Two  hundred 

Twelve 

ccc.   . 

Three  hundred 

Thirteen 

cccc  . 

Four  hundred 

Fourteen 

D      .     . 

Five  hundred 

Fifteen 

DC  .     . 

Six  hundred 

Sixteen 

DCC      . 

Seven  hundred 

Seventeen 

DCCC   . 

Eight  hundred 

Eighteen 

DCCCC 

Nine  hundred 

Nineteen 

M     .     . 

One  thousand 

LJrR     V.]  INTELLECTUAL    ARITHMETIC.  11 

LESSON    V. 
Addition  Table  from  1  to  3  inclusive. 

1.  One  and  one,  are  how  many*  1 

2.  One  and  two,  are  how  many  1 

3.  One  and  three,  are  how  many  1 

4.  One  and  four,  are  how  many  ? 

5.  One  and  five,  are  how  many  ? 

6.  One  and  six,  are  how  many  ? 

7.  One  and  seven,  are  how  many  1 

8.  One  and  eight,  are  how  many  ? 

9.  One  and  nine,  are  how  many  1 

10.  One  and  ten,  are  how  many  1 

11.  Two  and  one,  are  how  many  ? 

12.  Two  and  two,  are  how  many  1 

13.  Two  and  three,  are  how  many  ? 

14.  Two  and  four,  are  how  many? 

15.  Two  and  five,  are  how  many? 

16.  Two  and  six,  are  how  many  ? 

17.  Two  and  seven,  are  how  many  ? 

18.  Two  and  eight,  are  how  many  1 

19.  Two  and  nine,  are  how  many  ? 

20.  Two  and  ten,  are  how  many  ? 

21.  Three  and  one,  are  how  many  ? 

22.  Three  and  two,  are  how  many  ? 

23.  Three  and  three,  are  how  many  ? 

24.  Three  and  four,  are  how  many  1 

25.  Three  and  five,  are  how  many  ? 

26.  Three  and  six,  are  how  many  ? 

27.  Three  and  seven,  are  how  many? 

28.  Three  and  eight,  are  how  many  ? 

29.  Three  and  nine,  are  how  many  ? 

30.  Three  and  ten,  are  how  many  ? 

*  SUGGESTIONS. — The  sum  of  tAvo  or  more  numbers  contains  sjb 
man}'  units  us  there  are  in  the  numbers  added.  Thus,  2  is  the 
fum  of  1  and  1 ;  4  the  sum  of  2  and  2,  or  of  1  and  3.  » 

ADDITION  is  the  process  of  finding  the  sum  of  two  or  more 
numbers.    The  sign    +   (plus,)  placed  between  two 
signifies  that  they  are  to  be  added. 


1%  INTELLECTUAL    ARITHMETIC.  [SEC,  I. 

QUESTIONS. 

1.  How  many  fingers  have  you  on  one  hand,  not 
counting  the  thumb  ?     How  many  on  both  hands  1 

2.  Counting  the  thumb,  how  many  have  you  ou 
each  hand  ]     How  many  on  both  1 

3.  One  and  four  are  bow  many  ?     One  and  five  ] 
One  and  nine  1     One  and  ten  ? 

4.  James  has  one  apple  and  buys  five  :  how  many 
will  he  then  have  1 

5.  John  has  six  apples  and  buys  one  :  how  many 
will  he  then  have? 

6.  Charles  has  nine  marbles  and  John  gives  him 
one  :  how  many  will  he  then  have  ? 

7.  How  many  are  2  and  2  ]     How  many  are  2  and 
4  ?     2  and  3  ?     2  and  6  ? 

8.  How  many  are  2  anct  7  ?     How  many  are  2 
and  8  ?     How  many  are  2  and  9  ? 

9.  James  has  two  tops  and  buys  four :  how  many 
will  he  then  have  ?     Two  and  four  are  how  many  ? 

10.  John   has  two  apples  and  William  gives  him 
six  :  how  many  will  he  then  have  1 

11.  Bought  two  quills  for  two  cents,  and  four  quills 
for  four  cents  :  how  many  quills  did  1  buy  1 

12.  James  bought  2  apples  for  two  cents  and   8 
more  for  eight  cents :  how  many  did  he  buy  in  all  'I 
Three  arid  six  are  how  many  1 

13.  If  you  buy  two  peaches  for  two  cents  and  9 
peaches   for  nine  cents,  how   many  peaches  do   you 
buy  ]     How  much  do  you  pay  for  them  1 

14.  How  many  are  3  and  3  ?      How  many  are  3 
and  4  ? 

15.  John  has  three  nuts  in  one  hand  and  five  in  the 
other  :  how  many  in  both  ?     3  and  8  are  how  many  ? 

16.  James  has  three  pencils  and  John  five  :  how 
\»any  have  both  ?     3  and  six  are  how  many  1 


LE8.  VI,  J  INTELLECTUAL    ARITHMETIC.  13 

LESSON    VI. 
Addition  Table  from  4  to  6  inclusive 

1.  Four  and  one  are  how  many  1 

2.  Four  and  two  are  how  manyl 

3.  Four  and  three  are  how  many  ? 

4.  Four  and  four  are  how  many  ? 

5.  Four  and  five  are  how  many  1  , 

6.  Four  and  six  are  how  many  ? 

7.  Four  and  seven  are  how  many  ? 

8.  Four  and  eight  are  how  many  ? 

9.  Four  and  nine  are  how  many  1 

10.  Four  and  ten  are  how  many  1 

11.  Five  and  one  are  how  many  ? 

12.  Five  and  two  are  how  many  ? 

13.  Five  and  three  and  how  many  1 

14.  Five  and  four  are  how  many  1 

15.  Five  and  five  are  how  many  1 

16.  Five  and  six  are  how  many  1 

17.  Five  and  seven  are  how  many  ? 

18.  Five  and  eight  are  how  many  ? 

19.  Five  and  nine  are  how  many  ? 

20.  Five  and  ten  are  how  many  ? 

21.  Six  and  one  are  how  many  ] 

22.  Six  and  two  are  how  many  1 

23.  Six  and  three  are  how  many  1 

24.  Six  and  four  are  how  many  ] 

25.  Six  and  five  are  how  many  ? 

26.  Six  and  six  are  how  many  1 

27.  Six  and  seven  are  how  many  ? 
23.  Six  and  eight  are  how  many '? 

29.  Six  and  nine  are  how  many  ? 

30.  Six  and  ten  are  how  many  ] 

QUESTIONS. 

1.  John  has  four  tops  and  Charles  one  :  how  many, 
have  both  ? 

2 


14  INTELLECTUAL    ARITHMETIC'.  [s3C.    I, 

2.  William  has  four  apples  and  James  three  :  how 
many  have  both? 

3.  How  many  are  4  and  4?     How  many*  are  4 
and  5  ? 

4.  John  has  four  chestnuts  in  one  hand  and  three 
i.  the  other  :   how  many  has  he  in  both  ? 

5.  Charles  has  four  quills   and  John  seven  •  how 
jinny  have  both  ?     Four  and  7  are  how  many  ? 

6.  John    and    James    have    each    four  tops:    how 
many  have  both  ?     Four  and  9  are  how  many  ? 

7.  William  has  four  birds  in  one  cage  and  seven 
in  another:   how  many  in  both? 

8.  Jane  has  four  pins   in  her  cushion  and  puts  in 
six  more  :   how  many  will  she  then  have  ? 

9.  Mary   has   four  needles   and   buys  eight:  how 
many    will    she   then     have?      4    and    10    are    how 
many  ? 

10.  John  buys  three  pear.s  for  four  cents  and  six 
pears  for  eight  cents :  how  many  pears  does  he  buy  ? 

11.  How  many  are  5  and  1?     How  many  are  6 
and  3  ?     5  raid  8  ?     5  and  9  ?     5  and  10  ? 

12.  John  has  five  marbles  in  one  hand  and  eight  in 
the  other  :   how  many  in  both? 

13.  Charles  has  five  cents  and  his  father  gives  him 
seven  :  how  many  has  he  then  ? 

14.  John  has  five  apples  and  Reuben  gives  him 
uine  :   how  many  has  he  then? 

15.  Isaac  buys  five  sheets  of  paper  for  five  cents, 
mid  ten  sheets  more  for  ten  cents  :  how  many  sheets 
does  he  buy  ? 

10.  If  1  buy  five  oranges  for  five  cents,  and  six 
uianges  for  six  cents,  how  many  do  1  buy  ?  Five  and 
5  are  how  many  ? 

17.  How  many  are  G  and  1  ?  How  many  are  C 
2  ?  6  and  6  ?  6  and  8  ?  6  and  10  ? 


LES    VII.]  INTELLECTUAL    ARITHMETIC.  15 

18.  How  many  are  6  and  6?     How  many  are  6 
and  SI     6  and  9  1     6  and  2 ?     6  and  10 1 

19.  William  carries   six  apples   to   school  in  his 
basket  and  Henry  four  :  how  many  in  both  baskets  ? 

20.  John  has  six  apples,  and  hijj  sister  Jane  gives 
him  five:  how  many  has  he  then  ? 

21.  Charles  has  six  apples  and  wins   eight  from 
John  :  how  many  has  he  then? 

22.  William  buys  three  tops  for  dix  cents  and  eight 
tops  for   ten   cents:  how   many   t^ps  does  he  buy  ? 

23.  James   buys   six  eggs  for  b^x  cents  and  eight 
eggs  for  nine  cents  :  how  many  eg^o  does  he  buy  ? 

24.  Jane  has  6  apples  and  Mary  gives  her  9  :  how 
many  will  she  then  have  ? 


LESSON   VII. 

Addition  Table  from  7  to  9  inclusive. 

1.  Seven  and  one  are  how  many  1 

2.  Seven  and  two  are  how  many  ? 

3.  Seven  and  three  are  how  many'? 

4.  Seven  and  four  are  how  maify  ? 

5.  Seven  and  five  ar«  how  many  ? 

6.  Seven  and  six  are  how  many  ? 

7.  Seven  and  seven  are  how  many  ? 

8.  Seven  and  eight  are  how  many  ? 

9.  Seven  and  nine  are  how  many  ? 

10.  Seven  and  ten  are  how  many  ? 

11.  Eight  and  one  are  how  many? 

12.  Eight  and  two  are  how  many  1 

13.  Eight  and  three  ars  how  many] 

14.  Eight  and  four  are  how  many? 

15.  Eight  and  five  are  how  many? 

16.  Eight  and  six  are  how  many? 


16      '  INTELLECTUAL    ARITHMETIC.  JSEC.  I. 

17.  Eight  and  seven  are  how  many  ? 

18.  Eight  and  eight  are  how  many  ? 

19.  Eight  and  nine  are  how  many  ? 

20.  Eight  and  ten  are  how  many  1 

21.  Nine  and  one  are  how  many  ? 

22.  Nine  and  two  are  how  many  ? 

23.  Nine  and  three  are  how  many  ? 

24.  Nine  and  four  are  how  many  1 

25.  Nine  and  five  are  how  many  1 

26.  Nine  and  six  are  how  many  \ 

27.  Nine  and  seven  are  how  many  1 

28.  Nine  and  eight  are  how  many  1 

29.  Nine  and  nine  are  how  many  ? 

30.  Nine  and  ten  are  how  many  ? 

QUESTIONS. 

1.  How  many  are  7  and  1  ?     Plow  many  are  7  and 
2  ?     7  and  5 1   *7  and  4  '?     7  and  6  ? 

2.  James  has  seven  oranges  in  one  basket  and  six 
in  another:  how  many  in  both? 

3.  William  has  seven  apples  and  John  gives  him 
nine  :  how  many  has  he  then  '? 

4.  A  father  ha*, two  sons  and  gives  seven  cents  to 
each  :  how  many  cents  does  he  give  to  both  ? 

5.  If  Henry   buys  seven  apples,  and  Mary  gives 
him  nine  :  how  many  will  he  then  have  ? 

(>.  If  George  buys  seven  quills  at  one  time  and  8 
at  another  :  how  many  does  he  buy  in  all  ? 

7.  William  has  5  marbles  and  Henry  gives  him  8: 
how  many  will  he  then  have  ? 

8  How  many  are  8  and  2  ?  How  many  are  8  and 
4  ?  8  and  6  ?  8  and  5  ?  8  and  9  ? 

9.  A  boy  has  eight  marbles  and  gains  five :  how 
many  has  he  then?     8  and  10  are  how  many? 

10.  If  he  has  eight  and  gains  nine,  how  many  will 
tfe  have  ?     6  and  4  and  5,  are  how  many  ? 


fcES.  VII.]  INTELLECTUAL    ARITHMETIC. 

11.  If  George  buys  eight  marbles  for  three  cents 
and  eight  more  for  four  cents,  how  many  will  he  buy 
in  all  ? 

12.  John  has  eight  marbles  and  Charles  gives  him 
nine  :  how  many  has  he  then  ? 

13.  Eight   and  four  are  how  many?     Eight  and 
seven  how  many  ]     Eight  and  6  are  how  many  ? 

14.  How  many  are  9  and  2?     How  many  are  9 
and  4  1     1  and  3  and  6,  are  how  many  1 

15.  Charles  has  nine  apples  and  buys  five  more: 
how  many  has  he  then  1     9  and  9,  are  how  many  1 

16.  If  he  has  nine  and  buys  eight,  how  many  will 
he  tyave  rl     4  and  5  and  6,  are  how  many '? 

17.  Nine  and  seven  are  how  many  ]     8  and  7  ] 

18.  If  James  buys  nine  oranges  for  nine  cents  and 
eight  more  for  9  cents,  how  many  will  he  buy  in  all  ? 

19.  Six  sheets  of  paper  cost  nine  cents  and  2  pencils 
cost  one  cent:  what  do  the  paper  and  pencils  cost? 
6  and  7  and  8,  are  how  many ? 

20.  How  many  are  10  and  2?     How  many  are  10 
and  3  ?     10  and  9  T   8  and  7  and  6  1 

21.  James  has  ten  pencils  and  then  buys  eight: 
how  many  has  he  then  ?     9  and  4  and  3,  are  how 
many  ] 

22.  John  gives  ten  chestnuts  to  Henry  and  nine  to 
William  :  how  many  does  he  give  to  both  ? 

23.  James  spends  six  cents  for  candy,  four  cents 
for  gingerbread,  and  eight  cents  for  nuts :  how  much 
does  he  spend  in  all  ]     1  and  2  and  4  and  10,  are  ho\v 
many  1 

24.  Nancy  has  ten  pins  on  her  cushion,  and  sticks 
uine  more  there  :  how  many  will  she  then  have  1 

25.  Jane  has  ten  needles  and  Lucy  gives  her  seven : 
flow  many  will  she  then  have  ? 

26.  Oliver  buys  ten  oranges  for  twelve  cents  and 
ten  more  for  eight  cents  :  how  many  does  he  buy  1   , 

2* 


INTELLECTUAL    ARITHMETIC.  [SEC.  X. 

LESSON   VIII. 
Numbers  added  viith  reference  to  Ten. 
L  Five  and  5  are  how  many  ?     4  and  6  ?     3  and 
7  ?     2  and  8  ?     1  and  9  1* 

2.  Ten  and  10  are  how  many?     15  and  5  are  how 
many  ?     16  and  4  ?     17  and  3  ?     18  and  2  ?     19  and 

?     14  and  6?     13  and  7?     12  and  8?     11  and  9? 

3.  Twenty  and   ten  are  how  many  ?     25  and  5  'I 
26  and  4  ?     27  and  3  ?     28  and  2  ?     29  and  1  ?     24 
and  6  ?     23  and  7  ?     22  and  8  ?     21  and  9  ? 

4.  Thirty  and  10  are  how  many?     35  and  5?     36 
and  4  ?     37  and  3  ?     38  and  2  ?     39  and  1  ?     34  and 
6  ?     33  and  7  ?     32  and  8  ?     31  and  9  ? 

5.  Forty  and  10  are  how  many  ?     45  and  5  ?     46 
and  4  ?     47  and  3  ?     48  and  2  ?     49  and  1  ?     44  and 
6  ?     43  and  7  ?     42  and  8  ?     41  and  9  ? 

6.  Fifty  and  10  are  how  many?     55  and  5?     56 
and  4  ?     57  and  3  ?     58  and  2  ?     59  and  1  ?     54  and 
6?     53  and  7 ?     52  and  8 ?     51  and  9  ? 

7.  Sixty  and  10  are  how  many?     65  and  5?     66 
and  4  ?     67  and  3  ?     68  and  2  ?     69  and  1  ?     64  and 
6  ?     63  and  7  ?     62  and  8?     61  and  9  ? 

8.  Seventy  and  10  are  how  many?     75  and  5? 
76  and  4  ?     77  and  3  ?     78  and  2  ?     79  and  1  ? 

9.  Eighty  and  10  are  how  many  ?     85  and  5  ?     8V 
and  1  ?     87  and  3  ?     4  and  86  ?     2  and  88  ?     6  and 
84  ?     7  and  83  ?     9  and  81  ?     8  and  82  ? 

10.  Ninety  and  ten  are  how   many?  >  99  and  1  ? 
92  and  8  ?     7  and  93  ?     3  and  97  ?     5  and  95  ?     6 
and  94  ?     9  and  91  ?     98  and  2  ?     4  and  96  ? 

*SUGGESTIONS. — The  first  part  of  this  lesson  points  out  to  th« 

pupil  the  method  of  adding,  by  considering  what  numbers  make 

exact  tens.    This  is  very  important,  and  should  be  much  dwelt  on. 

The  second  part  shows  that  the  first  figure  of  a  Bum  is  always 

derived  from  the  units. 

«  The  questions  of  the  lesson  should  first  be  put  in  the  order  in 
which  they  ire  written,  and  then  promiscuously,  until  thorough 
ly,  learned 


LE8.    IX.]  INTELLECTUAL    ARITHMETIC. 

11.  Two  and  two  are   how  many?      12  and  2? 
22  and  2  ?     32  and  2 1     42  and  2  ?     52  and  2  ?     62 
and  2  ?     72  and  2  ?    82  and  2 1     92  and  2  ?     94  and 
2?     96  and  2?     98  arid  2? 

12.  Three  and  3  are  how  many  ?     13  and  3  ?     23 
and  3  ?     33  and  3  ?     43  and  3  ?     53  and  3  1     63  and 

?     73  and  3  ?     83  and  3  ?     93  and  3  ?     96  and  4  ? 

13.  Four  and  4  are  how  many  ?     4  and  14?     24 
and  4  ?     34  and  4  1     44  and  4  ?  *  54  and  4  ?     64  and 

4  ?     74  and  4  ?     84  and  4  ?     94  and  4  1     98  and  2  7 

14.  Five  and  5  are  how  many  ?     15  and  5  ?     25 
and  5  ?     35  and  5  ?     45  and  5  ?     55  and  5  ?     65  and 

5  ?     75  and  5  ?     85  and  5  ?     95  and  5  ? 

15.  Six  and  6  are  how  many  ?     16  and  6  ?    26  and 

6  ?     36  and  6.1     46  and  6  ?     66  and  6  ?     76  and  6  1 
86  and  6 ?     96  and  6  ? 

16.  Seven  and  7  are  how  many  ?     17  and  7  ?     27 
and  7?     37  and  7  ?     47  and  7  ?     57  and  7  ?     67  and 
7?     77  and  7?     87  and  7?     97  and  7  ? 

17.  Eight  and  8  are  how  many  1     18  and  8?     28 
and  8?     38  and  8?     48  and   8?     58  and   8?     68 
and  eight?     78  and  8  ?     88  and  8  ?     98  and  8  ? 

18.  Nine  and  9  are  how  many  ?     19  and  9?     29 
and  9  ?     39  and  9  ?     49  and  9  ?     59  and  9  ?    69  and 
9  ?     79  and  9  ?     89  and  9  ?     99  and  9  ? 

LESSON    IX 

Showing  the  formation  of  Numbers  from  11  to  100. 

1.  Eleven  and  1  are  how  many  ?  Eleven  and  2? 

Eleven  and  3  ?  Eleven  and  4  ?  Eleven  and  5  ? 

Eleven  and  6?   Eleven  and  7?   Eleven  and  8? 

Eleven  and  9  ?  Eleven  and  10  ?  Eleven  and  11  ?* 

*  SUGGESTION. — This  lesson  indicates  how  all  the  numbers  may 
be  formed  from  11  to  include  one  hundred  and  nine.  After  the 
questions  have  beeu  put  in  the  order  in  which  they 
ihey  should  be  put  promiscuously. 


ARITHMETICAL    ARITHMETIC.  I  SEC.  L 

V^  m  u 

2.  Twenty -two  and   1  are  how  many  ?     Twenty- 
two  &nd  2  ?     Twenty-two  and  3  ?     Twenty-two  and 
4  ?   Twenty-two  and  5  1    Twenty-two  and  (j  ?    Twen 
ty-two  and   7?     Twenty-two  and   8?     Twenty -two 
and  9  ?      Twenty-two    and    10  ?      Twenty-two  and 
111 

3.  Thirty-three  and   1   are  how   many?      Thirty- 
three  and  2  1     Thirty -three  and  3  ?     Thirty-three  and 
4?    Thirty-three  and  5  ?    Thirty-three  and  6  ?  Thirty- 
three  and' 7  ?     Thirty-three  and  87     Thirty-three  and 
9?      Thirty-three  and  10?      Thirty-three    and    11? 

4.  Forty-four  and    1  are  how  many?     Forty-four 
and  2?      Forty-four    and    3?      Forty-four    and    4? 
Forty-four  and  5?     Forty-four  and  (5?     Forty -four 
and  7?    Forty-four  and  8?    Forty-four  and  9?    Forty- 
four  arid  10  1     Forty-four  and  1 1  ? 

5.  Fifty -live  and  1  are  how  many?     Fifty -five  and 
'2?     Fiftv-five  and  3?     Fifty-five  and  4?    'Fifty-five 

and  5  ?  'Fifty -five  and  6  ?  Fifty-five  and  7  ?  Fifty- 
live  and  8?  Fifty-five  ancl  9  ?  Fifty-five  and  10? 
Fifty-five  and  11? 

6.  Sixty-six  and  1  are  how  many?     Sixty-six  and 
2?     Sixty-six  and   3  ?     Sixty-six  and  4  ?     Sixty-six 
and  5  ?     Sixty -six  and  6  ?     Sixty-six  ancl  7  ?     Sixty- 
six   and   8?     Sixty-six  and  9?     Sixty-six   and    10? 
Sixty-six  and  11  ? 

7.  Seventy-seven  and  1  are  how  many  ?     Seventy- 
seven  and  2  ?     Seventy-seven  and  3  ?     Seventy-seven 
and  4?     Seventy-seven  and  5?     Seventy-seven  and 
0?     Seventy-seven  and  7?     Seventy-seven    and    8? 
Seventy-seven  and  9?     Seventy -seven  and  10?     Sev 
£i)ty-seven  and  11  ? 

8.  Eighty-eight  and  1   are   how   many?     Eighty- 
sight  and    2?      Eighty -eight  and  3?      Eighty-eight 
and  4  ?     Eighty-eight  and  5?     Eighty-eight  and  6? 
'Eighty  eight  and  7  ?     Eighty -eight  and  8  ?     Eighty 


LES.  X.J  INTELLECTUAL    ARITHMETIC. 

eight   and  9?     Eighty-eight  and   10]     Eighty-eight 
and  11? 

9.  Ninety-riin^  and  1  are  how  many?  Ninety- 
nine  and  2?  Ninety -nine  and  3?  Ninety -nine  and 
4?  Ninety-nine  and  5  ?  Ninety-nine  and  6  1  Nine 
ty-nine  and  7?  Ninety-nine  and  8]  Ninety-inn* 
and  9  ?  Ninety-nine  and  10  ? 


.       LESSON  X. 

Practical  Questions. 

1.  Let  each  of  the  following  combinations  be  given 
as  a  separate  example.     How  many  are 


10  and  20  and 
10  and  30  and 
10  and  40  and 
10  and  50  and 
20  and  30  and 
15  and  20  and 
25  and  15  and 
35  and  12  and 
40  and  60  and 


4? 
9? 
6? 
3? 
9? 
6? 
4? 
3? 
9? 


8  and  20  and  10  1 


40  and  50  and  6? 

6  and  12  and  30? 

7  and    15  and  70? 
9  and   14  and  60? 

13  and     7  and  14? 

19  and  11   and  16? 

21  and     9  and  13? 

30  and  40  and  10? 

36  and     4  and  19? 

38  and  12  and  16? 

S    Jane  has  13  pins  in  her  cushion  and  Mary  27  •. 
how  many  pins  have  both  ? 

3.  John   has   a   number  of  pears  :  he  gives  8  to 
William,  12  to  Charles,  9   to  James  and  has  Tleft: 
how  many  had  he  at  first  ? 

4.  There  are  4  bags  of  coffee  :  the  first  contains  16 
pounds,  the   second   14   pounds,  the  third   7  pounds, 
and  the  fourth  3  pounds  :  how  many  pounds  in  all 
the  bags  ? 

5.  A  former  has  4  pastures  containing  sheep.    In  the 
first  there  are  3  sheep,  in  the  second  there  are  6,  ifl 


INTELLECTUAL    ARITHMETIC.  [BEC.  L 

the  --hird  there  are  7,  and  in  the  fourth  there  are  8: 
ho\\  many  are  there  in  the  four  pastures  ? 

6.  James  gave  18  cents  for  a  squirrel,  82  cents  for 
a  cage,  and  15  cents  for  nuts  :  how  many  cents  did 
he  pay  in  all  ? 

A  man  bought  a  cow  for  25  dollars,  a  calf  for  5 
dollars,  3  lambs  for  8  dollars,  and  a  pig  for  2  dollars  • 
what  did  he  pay  for  all  ? 

8.  How  many  are  1  and  2  and  4  and  14  and  9? 

9.  How  many  are  3  and  4  and  10  and  5  aid  4 
and  5  ? 

10.  How  many  are  4  and  14  and  16  and  0  and  7 
and  8 1 

11.  How  many  are  15  and  13  and  12  and  4  and  9? 

12.  How  many  are  9  and  11  and  14  and  16  and 
171 

13.  How  many  are  1  and  2  and  4  and  3  and  6  ? 

14.  How  many  are  2  and  2  and  4  and  3  and  5  ? 

15.  How  many  are  6  and  4  and  4  and  3  and  3  ? 

16.  How  many  are  6  and  4  and  3  and  6  and  5  ? 

17.  How  many  are  7  and  7  and  4  and  2  and  6 1 

18.  How  many  are  9  and  2  and  8  and  7  and  5 
and  8  ? 

19.  A  lady  bought  some  tape  for  10  cents,  some 
fcr  18  cents,  a  comb  fur  22  cents,  and  a  pair  of 

scissors  for  30  cents :  how  much  did  she  pay  in  till  ? 

20.  A  farmer  has   15  sheep  in  one  lot,  25  in  an 
other,  and  30  in  his  barn-yard :  how  many  lias  he  in 

aiir 

21.  A  merchant  buys  26  barrels  of  flour  of  one 
miller,  30  of  another,  14  of  another,  and  36  of  another : 
how  many  barrels  does  he  buy  in  all  ? 

22.  A  man  bought  a  horse,  saddle  and  bridle;  for 
the  horse  he  gave  75  dollars,  for  the  saddle  25  dol 
lars,  and  7  dollars  for  the  bridle :  what  did  they  all 
rjost  him? 


LE8.  X.]  INTELLECTUAL    ARITHMETIC. 

23.  A  drover  bought  12  sheep  of  one  farmer,  30 
of  another,  18  of  another,  and  25   of  another  :  how 
many  did  he  buy  in  all  1 

24.  James  gave  nine  cents  to  a  beggar  woman,*!  1 
cents  to  a  beggar  man,  and  8  cents  to  a  beggar  girl : 
'  ow  much  did  he  give  in  all  ? 

25.  If  John  has  14  cents  in  one  pocket,  10  cents  in 
another,  6  cents  in  his  purse,  and  8- cents  in  his  hard, 
how  many  cents  has  he  in  all  ? 

26.  Charles  has  8  cents,  William  18,  Robert  4,  and 
Samuel  9 :  how  many  cents  have  they  all  ? 

27..  If  Lucius  gives  36  cents  for  a  pen-knife,  8  cents 
for  paper,  6  cents  for  quills,  and  7  cents  for  wafers, 
how  much  does  he  pay  in  all '? 

28.  James  buys  9  sticks  of  white  candy,  9  sticks 
of  red  candy,  2  of  brown   candy,  and  8  of  yellow . 
candy  :  how  many  sticks  does  he  buy  in  all  ? 

29.  If  James  gave  18  cents  for  the  white  candy,  18 
cents  for  the  red  candy,  4  cents  for  the  brown  candy, 
and  16  cents  for  the  yellow  candy,  how  much  did  he 
pay  in  all  ? 

30.  Jane  pays   18  cents  for  a  slate,  12  cents  for 
quills,  11  cents  for  paper,  and  15  cents  for  pencils  : 
what  does  she  pay  in  all  1 

31.  A  grocer  purchases  6  barrels  of  flour  for  30 
dollars,  a  load  of  hay  for  12  dollars,  and  10  bushels 
of  oats  for  15  dollars  :  how  much  did  he  pay  in  all  1 

32.  A  tailor  paid    15  dollars  for  a  piece  of  cloth, 
for  a  coat,  4  dollars   for  the  lining,  2  dollars  for  the 
buttons,  and  charged  9  dollars  for  making  :  what  was 

the  cost  of  the  coat?   £  /  } 

-tj     -«•  ***^     -.Jr ' 


^    INTELLECTUAL    ARITHMETIC.  [SEO.  EL 

SECTJON    SECOND. 


LESSON    I. 

Subtraction. 

1.  One  and    1   are  2  :  if  we  take   1  from  2  what 
remains*  ? 

2.  One   and  2  are  3  :  if  we  take  1  from  3.  whai 
remains  ?     If  \ve  take  2  from  3,  what  remains? 

3.  One   and  3  are  4  :  if  we  take   1   from  4,  what 
remains  ?     If  we  -take  3  from  4,  what  remains  1     If 
we  take  2  from  4,  what  remains? 

4.  One   and   4   are  5  :  if  we  take  1  from  5,  what 
remains?     5  less  4,   are   how   many  ?     5  less  2,  aie 
how  many  ?     5  less  3,  are  how  many  ? 

5.  One  and   5  are  .6  :    6   less   5,  are  how  many? 
6  less  1,  are  how  many  ?     6  less  2.  are  how  many  ? 
6  less  3.  are  how  many  ?     6  less  4,  are  how  many  ] 

6.  Seven  less  1  are  how  many  ?     7  less  2,  are  how 
many  ?     7  less  4,  are  how  many  ?     7  less  5,  are  Low 
many  ?     7  less  0,  are  how4  many  ?   . 

*  SUGGESTIONS. — The  pupil  first  gets  the  idea  of  'more  by 
addition.  He  adds  two  numbers  together  and  finds  that  their 
sum  is  greater  than  either  of  them.  He  next  sees  that  if  one 
of  them  be  taken  away,  the  other  will  be  left :  hence, 

The  difference  between  two  numbers  is  such  a  number  as  addtd 
to  the  lens  will  give  the  greater, 

SUBTRACTION  is  the  process  of  finding  the  difierence  between 
two  numbers. 

Teach  the  pupil  that  every  question  in  Subtraction  requires 
him  to  find  such  a  number  as  added  to  the  less  will  give  the 
greater. 

What  is  the  difference  between  6  and  2  ?  Why  is  4  the  dif 
ference  between  6  and  2  ?  Because  4  added  to  2  gives  6. 

The  sign  —  (minus),  placed  between  two  numbers  signifies 
that  the  cumber  after  it  is  to  be  taken  from  the  number  before 
it 


LES.  I.]          _       INTELLECTUAL    ARITHMETIC. 

7.  Eight  less  1,  are  how  many  1     8  less  3,  are  how 
many  ?     8  less  5,  are  how  many  ?     8  less  7,  are  how 
many  1 

8.  Nine  less  1,  are  how  many?     9  less  3,  are  how 
many  ?     9  less  5,  are  how  many  ? 

9.  Eight  and  2,  are  how  many  ?     10  less  2,  are 
how  many  ?     10  less  8,  are  how  many1? 

10.  Six  and  4,  are  how  many  ?     10  less  G,  are  how 
many?     11  less  5,  are  how  many  ? 

11.  William  has  three  apples  and  gives  them  all 
to  James,  how  many  has  he  left?     3  less  3,  what 
remains? 

12.  William   has  six  apples  and  gives    three    to 
J times:  how  many  has  he  left?     G  less  3,  are  how 
many  ? 

13.  7  less  3,  are  how  many? 

14.  9  less  3,  are  how  many  ? 

15.  €0  less  3,  are  how  many  ? 
lt>.   14  less  3,  are  how  many  ? 

17.  4  less  4,  what  remains  ? 

18.  8  less  4,  are  how  many  ? 

19.  Fourteen  less  4,  are  how  many-^ 

20.  Henry  has  five  pears  in  a  basket  and  gives 
them  all  to  his  sister:  how  many  has  he  left?     5  less 
5   what  remains? 

21.  8  less  5,  leaves  how  many  7 

22.  9  less  5,  leaves  how  many  ? 

23.  11  less  5,  leaves  how  many  ? 

24.  James  has  six  squirrels  in  a  cage,  and  takes 
them  all  out:  how  many  will  be  left?     G  from  ft, 
Leaves  how  many  ? 

^5.  9  less  G,  leaves  how  many  ? 
2G.  8  less  G,  are  how  many  ? 

27.  10  less  G,  are  how  many  ? 

28.  Mary  has  seven  pins  in  her  cushion  and  takefr 
them  all  out:  how  many  are  left?     7  less  7,  whal 
remains  1  3 


INTELLECTUAL    ARITHMETIC.  [SEC.    EL 

29.  12  less  7,  are  how  many  ? 

30.  15  less  7,  are  how  many  ? 

31.  Reuben  has  eight  plums  and  gives  them  all  to 
John:    how   many   has   he   left?     8    less    8,    what 
remains  ? 

32.  10  less  8,  are  how  many  ? 

33.  14  less  8,  are  haw  many  ? 

34.  18  less  8,  are  how  many  ? 

35.  There  are  nine  chairs  in  a  room,  and  Mary 
Cakes  them  all  out :  how  many  are  left  ?     9  less  0, 
what  remains  ? 

36.  12  less  9,  are  how  many  ? 

37.  15  less  9,  are  how  many  ? 

38.  19  less  9,  are  how  many  ? 

39.  16  less  6,  are  how  many? 

40.  15  less  10,  are  how  many  ? 

41.  14  less  4,  are  how  many  ? 

42.  25  less  5,  are  how  many  ? 

43.  36  less  16,  are  how  many? 

44.  25  less  12,  are  how  many? 

45.  30  less  7,  are  how  many  ? 
16.  20  less  14,  are  ty>w  many  ? 

47.  30  less  12?  are  how  many  ? 

48.  27  less  7,  are  how  many  ? 

49.  29  less  10,  are  how  many!., 

50.  39  less  20,  are  how  many  ? 

51.  42  less  12,  are  how  many? 

52.  19  less  7,  are  how  many  ? 

53.  50  less  20,  are  how  many  ? 

54.  14  less  8,  are  how  many? 

55.  15  less  3,  are  how  many? 

56.  24  less  4,  are  how  many  ? 

57.  16  less  5,  are  how  many  ? 

58.  17  less  8,  are  how  many  ? 

59.  19  less  7,  are  how  many  ? 
CO.  19  less  9,  are  how  many  ? 


LES.    I.]  INTELLECTITAL    ARITHMETIC. 

'61.  29  less  8,  are  how  many? 

62.  34  less  8,  are  how  many  ? 

63.  35  less  6,  are  how  many  ? 

64.  50  less  8,  are  how  many  ? 

65.  57  less  6,  are  how  many  ? 

66.  59  less  5,  are  how  many  1 

67.  53  less  7,  are  how  many1? 
.68.  60  less  20,  are  how  ninny  ? 

69.  65  less  15,  are  how  many  ? 

70.  67  less  10,  are  how  many  ? 

71.  67  less  9,  are  how  many'* 

72.  70  less  5,  are  how  many  ? 

73.  74  less  3,  are  how  many  ? 

74.  78  less  8,  are  how  many  1 

75.  79  less  10,  are  how  many  ]  ^ 

QUESTIONS. 

1.  There  are  nineteen  peach-trees    in  an  orchard, 
arid  six  of  them  are  blown  down  in  a  storm :  how 
many  are  left  standing  1 

2.  Laura  has  twenty-five  cents,  and  buys  an  arith 
metic  for  eighteen  cents  :  ho,w  much  money  will  she 
have  left? 

3.  There  are  thirty-four  pears  in  a  basket,  and  nine 
of  them  are  taken  out :  how  many  are  left  ? 

4.  There  are  sixty-five  pigeous  in  a  flock,  and  John 
fires  at  them  and  kills  nine  :  how  many  are  left  1 

5.  There  are  fifty-four  sheep  in  a  fold,  and  a  wolf 
breaks  in  and  kills  seven  :  how  many  are  left  ? 

6.  There  are  forty -nine  scholars  in  a  school,  and 
ten  of  them  are  girls  :  how  many  boys  are  there  ? 

7.  In  another  school  there  are  twenty  scholars,  and 
nine  are  boys :  how  many  girls  are  there  ? 

8.  In  Elizabeth's  flower-bed  there  are  thirty  beauti 
ful  lilies,  and  John  breaks  off*  seven  of  them :  how 
many  are  unbroken  ? 


INTELLECTUAL    ARITHMETIC.  [SEC.    I. 

^  James  has  thirty-seven  cents  :  he  spends  six  foi 
candy,  eight  for  a  pencil,  and  twelve  for  a  pen-knife : 
how  many  has  he  left  1 

10.  John  has  twenty -five  cents,  and  spends  six  cents 
for  a  top,  nine  cents  for  a  pencil,  and  two  cents  for  & 
peach :  how  much  has  he  left  ? 

11.  Wiliam  has  37  cents,  lie  buys  a  top  for   10 
cents,  and  8  marbles  for  2  cents :  how  much  has  he 
left? 

12.  A  boy  has  40  peaches  :  he  gives  24  to  Lucy 
and  9  to  Elizabeth  :  how  many  has  he  left  1  >>.^ 

13.  James  received  a  premium  worth   56  cents  ; 
Jane  received  one  worth  30  cents :  what  was  the  dif 
ference  of  their  values  1 

14.  Charles  has  49  cents,  and  buys  a  book  which 
costs  him  29  cents:  how  much  has  he  left? 

15.  A  butcher  buys  39  sheep,  and  kills   17:  how 
many  are  left  alive  ? 

16.  *A  grocer  has  a  tub  of  butter  containing  45 
pounds;   he  sells  20  pounds  to  Mr.  Wilson,  and  15. 
pounds  to  Mr.  Jones  :  how  much  is  left? 

17.  How  many  are  55  less  17? 

18.  How  many  are  77  less  19  I 

19.  Four  men  bought  a  horse  for  56  dollars;  the 
first  paid  16  dollars,  the  second  20  dollars,  and   the 
third  14  dollars  :  what  did  the  fourth  pay  ? 

20.  Charles  bought  a  penknife  for  48  cents,  and  a 
top  for  25  cents  :  how  much  more  did  he  pay  for  the 
knife  than  for  the  top  ?  ^ 


LESSON    II. 
Questions  involving  Addition  and  Substraction. 

1.  Six  and  4  and  3  and  5,  less  4,  are  how  many  ? 

2.  Eight  and  9  and  4  and  2,  less  3,  are  how  many  ?  ' 
8.  Seven  and  6  and^aud  8,  less  5,  are  how  many  ? 


LES.    II.]  INTELLECTUAL    ARITHMETIC. 

4.  Nine  and  2  and  3  and  7,  and  1,  less  6,  are  how 
many  ? 

5.  Thirty  and  5   and  8  and  15,  less  10,  are  how 
many  ? 

6.  Twenty-one    and  6  and    13,  less  12,  aie  how 
many  ? 

7.  ^Forty-five   and    15  and    12,   less  9,  are  how 
many  ? 

8.  Sixty-nine  and  11  and  5  and  2,  less  8,  are  how 
many  ? 

9.  Seventy-five  and  5  and  6  and  12,  less  8,  are 
how  many  1 

10.  Forty-five  and  8  and  4  and  3  and   6  and  7 
and  9,  less  12,  are  how  many  1 

11.  James  has  4o  cents,  and  pays  12  cents  for  a 
whistle  and  25  cents  for  a  knife  :  how  much  has  he 
left? 

12.  A  man  bought  a  calf  for  5  dollars,  a  sheep  for 
4  dollars,  and  a  pig  for  2  dollars  ;  also  a  cow  for  25 
dollars  :  what  did  he  pay  in  all,  and  how  much  more 
for  the  cow  than  for  the  other  animals  1 

13.  James  has  26  nuts  in  cttie  pocket,  and  14  less 
in  the  other  pocket:  how  many  has  he  in  both? 

14.  A  man  has  50  dollars  ;  he  pays  26  for  a  coat, 
8  dollars  for  a  pair  of  pantaloons,  and  4  dollars  for  a 
vest :  how  much  has  he  left  ? 

15.  A  school-boy  pays  56  cents  for  an  Atlas.  30 
cents  foi   an  Arithmetic,  and  24  cents  for  a  slate  : 
what  did  he  pay  for  all,  and  how  much  more  for  the 
Atlas  than  for  the  Arithmetic  and  slate  ? 

16.  Eighty-five  and  5   and  9  and  3  and  J2.  Jess 
8,  less  3,  are  how  many  ? 

17.  Fifty-nine  and  5  and  9  and  8   and  7  and  0, 
less  8,  are  how  many  1 

18.  Forty-seven  and  9  and  6  and  4  and  5  and  S, 
less  12,  are  how  many  ?  __ 

3* 


INTELLECTUAL    ARITHMETIC.  [SEC.  IL 

19.  Seventy-two  and   15  and  6,  less  1-1 ,  are  how 
many  ? 

20.  Thirty  eight  and  4  and  9  and  7  and  6  and  2, 
less  12,  are  how  many? 

21.  William  had  16  marbles,  James  gave  him  7, 
John  gave  him  8,  and   Reuben  gave   him   enough  to 
make    his    number,    40 :    how    many    marble*    did 
Reuben  give  John  ? 

22.  A  father  gave  seventeen  cents  to  Lucy,  13  to 
Mary,  and  4   to  Jane,  and  then   took  back  9   from 
Lucy  :  how  many  had  they  left? 

23.  A  man  travelled  49  miles  in  three  days  ;  the 
first  day  he  travelled   16  miles,  and  the  second  day 
13  miles :  how  far  did  he  travel  the  third  day  ? 

__  24.  A  tailor  has  a  piece  of  cloth  containing  39 
yards  ;  he  sold  14  yards  to  one  man,  13  to  another, 
and  made  a  coat  which  took  two  yards:  how  many 
yards  were  left  ? 

25.  A  merchant  bought  some  coffee,  for  which  he 
paid  25  dollars,   some  sugar,   for  which  he  paid   12 
dollars,  and  some  tea,  for  which  he  paid  1 1  dollars ; 
he  sold  the  whole  for  5|-  dollars  :  what  did  he  gain  ] 

26.  A  tailor  bought  a.  piece  of  cloth  for  45  dollars ; 
he  made  it  into  a  coat  and  pantaloons ;   he  paid  10 
dollars  for  making,  and  then  sold  them  for  60  dollars : 
did  he  make  or  lose,  and  how  much  ? 

27.  James  has  49  peaches;  he  gives  15  to  Robert, 
13  to  John,  and  9  to   William  :  how  many  has  he 
left? 

28.  Charles  has  some  pears,  and  gives  12  to  James, 
1 1  to  Henry,  13  to  Reuben,  and  8  to  Elisha  ;  when 
he  finds  that  he  has  9  left:  how  many  had  he  at 
first1?  4 

29.  Lucy  has  1    pins  on  one  cushion,  and   15  on 
another:  if  she  takes  off  9,  how  many  will  she  have 


LES.  II.]  INTELLECTUAL    ARITHMETIC. 

30.  A  man  travelled  5  miles  before  breakfast, 
miles  between  breakfast  and  dinner,  and  then  travelled 
back    12  miles:  how  far  was  he  from  the  place  of 
starting  1 

31.  A  -cow  has  two   calves,  the  first  is  worth   3 
dollars,  the   second  4  dollars,  and  the  cow  is  worth 
25  dollars :  how  much   more   is  the  cow  worth  than 
the  two  calves,  and  what  are  they  all  worth  ? 

32.  A  grocer   buys  some  lemons  for  15  dollars, 
some  oranges  for  25  dollars,  and  then  sold  the  whole 
for  56  dollars  :  how  much  did  he  make  1. 

33.  Jane  has  32  rose-buds  on  one  bush,  and  16  on 
another,  and  38  only  blossom :  how  many  buds  did 
not  flower  ? 

34.  A  man  owes  55  dollars ;  at  one  time  he  pays 
24  dollars,  at  another  17  dollars,  and  then  14  dollars  •:  , 
how  much  does  he  then  owe  ? 

35.  William  went  after  chestnuts;   he  put  26  in 
one  pocket,  15  in  another,  and  16  in  a  third  :  he  lost 
9   out  of  the  first  pocket,  3  from  the  second,  and  6 
from  the  third  :  how  many  had  he  left? 

36.  Twenty-nine  plus  6  plus  8  plus  4,  less  12,  less 
11,  are  how  many? 

37.  Mr.  Jones  owes  his  baker  17  dollars,  his  gro 
cer  16  dollars,  and  his  tailor  27  dollars :  how  much 
does  he  owe  in  all,  and  how  much  more  to  his  baker 
and  grocer  than  to  his  tailor? 

38.  A  farmer  has  30  sheep  in  one  lot,  25  in  ari- 
other^and  16  in  a  third:  9   sheep  escape  from  the 
first  lot,  12  from  the  second,  and  9  from  the   third  : 
how  many  sheep  had  he  in  all,  how  m/inv 

and  how  many  were  left  in  the  fields  2 


INTELLECTUAL    ARITHMETIC.  [SKO.  IIL 


SECTION    THIRD. 


Once    1 
Once    2 
Once    3 
Once    4 
Once    5 
Once    6 
2  times  I 
2  times  2 
2  times  3 
2  times  4 
2  times  5 

2  times  6 

3  times  1 
3  times  2 
3  times  3 
3  times  4 
3  times  5 

3  times  6 

4  times  1 
4  times  2 
4  times  3 
4  times  4 
4  times  5 
4  times  6 


Once 

7 

is 

how 

many  ? 

Once 

8 

is 

how 

many  1 

Once 

9 

is 

how 

many  ? 

Once 

10 

is 

how 

many  ? 

Once 

11 

is 

how 

many? 

Once 

12 

is 

how 

many  ? 

LESSON    I. 

Multipliers  from  1  to  4  inclusive. 
is  how  many  ?* 
is  how  many  1 
is  how  many  ? 
is  how  many  ? 
is  how  many  1 
is  how  many  ? 
are  how  many  ?  2  times  7  are  how  many  ? 
are  how  many  ?  2  times  8  are  how  many  ? 
are  how  many  ?  2  times  9  are  how  many  ? 
are  how  many  ?  2  times  10  are  how  many  1 
are  how  many  ?  j  2  times  1 1  are  how  many  ? 
are  how  many  1  \  2  times  12  are  how  'many  ? 
are  how  many  ?  ,  3  times  7  are  how  many  ? 
are  how  many  ?  3  times  8  are  how  many? 
are  how  many  1  '3  times  9  are  how  many? 
are  how  many  ?  j  3  times  10  are  how  many  1 
are  how  many  1 . 3  times  11  are  how  many  ? 
are  how  many  1  \  3  times  12  are  how  many  ? 


are  how  many  1 
are  how  many  1 
are  how  many  ? 


4  times  7  are  how  many  ? 
4  times  8  are  how  many  ? 
4  times  9  are  how  many  ? 


are  how  many  1  4  times  10  are  how  many  ? 
are  how  many  1  4  times  11  are  how  many  ? 
are  how  many  ?  1 4  times  12  are  how  many  ? 


*  MULTIPLICATION  is  a  short  process  of  taking  one  number  as 
many  times  as  there  are  units  in  anotfier. 

The  number  to  be  taken  is  called  the  multiplicand.  The 
camber  denoting  how  many  times  the  'multiplicand  is  to  be 
taken,  is  called  the  multiplier. 

The  result  or  answer,  is  called  the  product 

Ilie  multiplicand  and  multiplier  are  called  factors  or  pro- 
facer*  of  the  product 


LES,  II.  J  INTELLECTUAL    ARITHMETIC.  33 

1.  If  James  buys  2  oranges  at  3  cents  apiece,  what 
do  they  cost?* 

2.  What  will  4  peaches  cost  at  1  cent  each  ? 

3.  What  will  3  oranges  cost  at  3  cents  apiece  ? 

4.  What  will  4  lemons  cost  at  4  cents  apiece  ? 

5.  What  will  3  pounds  of  raisins  cost  at  12  cents 
*  pound  ? 

6.  What  will  be  the  cost  of  4  tops  at  12  cents  apiece? 

7.  What  will  be  the  cost  of  2  melons   at  11  cents 
apiece  ? 


LESSON    II. 
Multipliers  from  6  to  8  inclusive. 

5  times  1  are  how  many  ?  •  5  times*  7  are  how  many? 
5  times  2  are  how  many^?  j  5/times  8  are  how  many  ? 
5  times  3  are  how  mai^y*?  j  Sp'times  9  are  how  many  ? 
5  times  4  are  how  many  ?  5  times  10 are  how  many? 
5  times  5  are  how  many  ?  ,  5  times  1 1  are  how  many  ? 

5  times  6  are  how  many  ?  j  5  times  12  are  how  many  ? 

6  times  1  are  how  many  ?   6  times    7  are  how  many  ? 

6  times    8  are  how  many  ? 


6  times    9  are  how  many  ? 


6  times  2  are  how  many  ? 

6  times  3  are  how 

6  times  4  are  how  many  ?  ;  G  times  10  are  how  many  ?' 

6  times  5  are  how  many  ?  G  times  1 1  are  how  many  ? 

6  times  G  are  how  many  ?  G  times  12  are  how  many  ? 

7  times  1  are  how  many  ?  |7  times    7  are  how  riij 
7  times  2  are  how  many  ?  7  times    8  are  how  in* 

7  times  3  are  how  many  ?|7  times  Dare  how  many  ' 
7  times  4  are  how  many  ?  7  times  10  are  how  many? 
7  times  5  are  how  many  ?  7  times  1 1  are  how  many  ] 


7  times  6  are  how  many  ? 


7  times  12  are  how  ruanv  } 


*  ANALYSIS. — Two  oranges  will  cost  two  times  as  mu»b  aa 
one  orange.     Since  1  orange  costs  8  cents  2  oranges  will  e< 
two  times  3  cents,  which  are  6  cents  :  therefore,  2  orange 
esctft  apiece,  will  cost  6  cents. 

2 


84  INTELLECTUAL    ARITHMETIC.  [SEC. 


8  times  1  are  how  many  ? 
8  times  2  are  how  many  ? 
8  times  3  are  how  many  ? 


8  times  7  are  how  many  ? 
8  times  8  are  how  many  ? 
8  times  9  are' how  many  1 


8  times  4  are  how  many  ?  j  8  times  10  are  how  many  1 
8  times  5  are  how  many  ?  !  8  times  11  are  how  many  ? 
S  times  G  are  how  many  ?  j  8  times  12  are  how  many  ? 

1.  If  William  buys  5  pine-apples  at  4  cents  each, 
what  do  they  cost  him? 

2.  What  is  the  cost  .of  6  barrels  of  flour  at  5  dol 
lars  a  barrel  1 

3.  What  is  the  cost  of  7  yards  of  cloth  at  9  dollars 
a  yard  1 

4.  What  is  the  cost  of  8  barrels  of  fish  at  8  dollars 
a  barrel?  \ 

5.  What  is  the  cosfe%  ofj6  pounds  of  candles  at  S 
cents  a  pound  ?  t      f 

6.  What  is  the  cost  of  8  pounds  of  raisins  at  11 
cents  a  pound  1 

7.  What  is  the  cost  of  7  pounds  of  sugar  at  8  cents 
a  pound  1 

8.  What  is  the  cost  of  5  pounds  of  beef  at  1 1  cents 
a  pound  1 


9  tin 


LESSON   fll. 
Multipliers  from  9  to  12  inclusive. 


times  1  are  how  many  1 
9  times  2  are  how  u.any  ? 
9  times  3  are  how  many  1 
9  times  4  are  how  many  ? 
9  times  5  are  how  many  1 
9  times  6  are  how  many ? 


9  times  7  are  how  many  1 
9  times  8  are  how  many  ? 
9  times  9  are  how  many  1 
9  times  10  are  how  many  ? 
9  times  11  are  how  many  ? 
9  times  12  are  how  manj  I 


NOTE. — The  sign  of  multiplication  (X),  placed  between  1wo 
sr  more  numbers,  signifies  that  they  are  k  be  multiplied 
together. 


LES.   III.J  INTELLECTUAL    ARITHMETIC. 


35 


10  times  2  are  how  many  ? 
10  times  3  are  how  many  ? 
10  times  4  are  how  many  ? 
10  times  5  are  how  many  1 

10  times  6  are  how  many  ? 

1 1  times  1  are  how  many  1 
11  times  2  are  how  many  ? 
1 1  times  3  are  how  many  1 
1 1  times  4  are  how  many  ? 
1 1  times  5  are  how  many  ? 

1 1  times  6  are  how  many  1 

12  times  1  are  how  many  1 
12  times  2  are  how  many  ? 
12  times  3  are  how  many  1 
12  times  4  are  how  many  1 
12  times  5  are  how  many  1 
12  times  6  are  how  many  ? 


times 

1 0  times  8  areliow  many  f 
10  times  9  are  how  many? 
10  times  10  are  how  many  ? 
10  times  11  are  how  many  ? 

10  times  12  are  how  many 

1 1  ti  m  es    7  are  how  m  any 

]  1  times  8  are  how  many  ? 
1 1  times  9  are  how  many  ? 
1 1  times  10  are  how  many  ? 
1 1  times  1 1  are  how  many  ? 

1 1  times  12  are  how  many  ? 

12  times    7  are  how  many? 
12  ti  m  es    8  are  how  many  ? 
12  times    9  are  how  many? 
12  times  10  are  how  many  ? 
12  times  11  are  how  many  ? 
12  times  12  are  how  many  ? 


1.  If  James  buys  9  lemons  at  2  cents  each,  what 
do  they  cost  him  ? 

2.  What  will  11  yards  of  calico  cost  at  11  cents  a 
yard? 

3.  What  will  12  dozen  of  apples  cost  at  12  cents 
a  dozen  ? 

4.  What  will  9  pine-apples  cost  at  11  cents  apiece? 

5.  What  will  12  yards  of  cloth  cost  at  9  dollars  a 
yard  ? 

6.  What  will  9  pumpkins  cost  at  12  cents  apiece? 

7.  What  will  be  the  cost  of  11  pairs  of  boots  at 
5  dollars  a  pair? 

8.  What  will  be  the  cost  of  12  loaves  of  bread  at 
11  cents  a  loaf? 

9    What  will  9  slates  cost  at  11  cents  apiece? 
10.   What  will  be  the  cost  of  11  yards  of  brood 
cloth  at  12  dollars  a  yard  ? 


SO  INTELLECTUAL    ARITHMETIC.  [j&KC.  IIL 

LESSON    IV. 

QUESTIONS. 

1.  What  is  the  product  of  13  taken  2  times]* 

2.  What  is  the  product  of  14  taken  3  times  ? 
fc  3.    What  is  the  product  of  15  taken  5  times? 

4.  What   is  the  product  of   18    taken    6    times 
I)  times?     7  times? 

5.  What   is   the  product    of   16  taken    4    times] 
5  times  ]     8  times  ] 

G.  If  James  reads  16  verses  of  the  bible  a  day, 
how  many  verses  will  he  read  in  a  week] 

7.  How  many  are  7  times  15?     Which  the  multi 
plicand?     Which   the  multiplier]     Which  the  pro 
duct  ? 

8.  What  will  9  pounds  of  butter  cost  at  19  cents  a 
pound  ? 

9.  If  a  steamboat  can  go  16  miles  an  hour,  how 
far  can  it  go  in  11  hours] 

10  How  many  are  2  times  20]  4  times  20? 
5  times  20?  7  times  20?  9  times  20?  11  times 
20?  12  times  20? 

11.  If  1  barrel  of  flour  cost  9  dollars,  what  will  12 
barrels  cost? 

12.  If  1  barrel  of  fUh  cost  7  dollars,  what  will   be 
the  cost  of  9  barrels? 

13.  If  1  yard  of  cloth  cost  6  dollars,  what  will  be 
the  cost  of  8  yards  ]      . 

14.  If  1  yard  of  cloth  costs  5  dollars,  what  will  15 
yards  cost] 

15.  If  a  man  on  horseback  can  ride  8  miles  in  1 
huur,  how  far  can  he  ride  in  12  hours? 

*  ANALYBIS. — Thirteen  is  made  up  of  1  ten  and  3  units.  Then 
1  ten  multiplied  by  2  gives  2  tens,  or  20;  and  8  units  multiplied 
ly  2  gives  6  uniis,  which  being  added  to  20  gives  2ti  tor  the 
product.  Let  each  question  be  analyzed  in  a  similar  manner. 


LKS.   V.j  INTELLECTUAL    ARITHMETIC.  37 

16.  How    many    are   2  times  18 1     3  times  18  ? 
5  times  r£ 

17.  ITow  many  are  p  times  17?      3  times  18? 
4  times  ? 

18.  If  1  cow  cos.t  20  dole's,  how  much  will  7  cows 
cost?     How  much  will  9  cost?     How  much  will  12 
cost? 

19.  If  a  man  travels  19  miles  a  day,  how  far  will 
he  travel  in  9  days  ?     In  10  days  ?     In  11  days  ?     In 
12  days  ? 

20.  11  a  pound  of  raisins  cost  15  cents,  how  much 
will  5  pounds  cost!  6  pounds?  7  pounds?    11  pounds? 

21.  If  a  man  digs  18  bushels  of  potatoes  in  1  day, 
how   many  would  he   dig  in   7  days?     In   10  days  I 
In  12  days? 

22.  If  a  man  eat  15  ounces  of  bread  in  a  day,  how 
many  ounces  will  he  eat  in  a  week  ?     How  many  in 
0  dajs  ?     In  11  days. 


LESSON  V. 
A  ddition. — Subtract  ion. — Multiplication. 

1.  What   will   be  the  cost  of  5  oranges  at  3  cents 
apiece  \ 

2.  What  will  7  books  cost  at  8  cents  apiece  ?     At 
9  cents?      At  15  cents  ?      At  20  cents? 

3.  What  will  be  the  cost  of  6  loaves  of  bread  at  9 
cents  a  loaf  ?     Of  8  loaves  ?     Of  10  loaves  ?     Of  11 
loaves?     Of  12  loaves'? 

4.  What  will  be  the  cost  of  4  hats  at  7  dollars 
apiece,  and  the  cost  <»f  2  coats  at  12  dollars  apiece? 

5.  What  will  be  the  cost  of  1 2  chickens  at  12  cent 
apiece  ?     Of  11?     Of  9  ?     Of  7  ?     Of  6  ?     Of  5  ? 

6.  What  will  be  the  difference  in  cost  between  2 
pencils  at  9  cents  apiece,  and  3  tops  at  4  cents  apiece  1 
How  much  would  they  all  cost? 


V 


38  INTELLECTUAL    ARITHMETIC.  [SEC.  III. 

7.  What  will  b^  the  cost  of  8  yards  of  ribbon  at 
8  cents  a  yard  ?     Df  5  yards  1     Of  G  yard£?     Of  9 
yards?     0£  12  yards]  f 

8.  What  is  the  difference  between  the  cost  of  8 
yards  of  cloth  at  5  dolla^a  yard,  .and.  12  yards  at  2 
lollars  a  yard?     What  would   be  the  cost  of  the 
whole? 

9.  James  has  25  marbles  and  gives  9  to  William, 
then  Charles  gives  him  as  many  as  he  has  left :  how 
many  more  than  he  had  at  first  will  he  then  have  ? 

10.  What  will  be  the  whole  cost  of  8  apples  at 

I  cent  apiece,  and  of  4  pears  at  2  cents  apiece? 

11.  What  will  be  the  whole  cost  of  6  oranges  at 
4  cents  apiece,  and  of  2  lemons  at  3  cents  apiece? 

12.  What  will  be  the  whole  cost  of  6  lemons  at  2 
cents  apiece,  and  of  2  apples  at  1  cent  apiece  ? 

13.  What  will  he  the  whole  cost  of  8  quills  at  2 
cents  apiece,  and  of  12  sheets  of  paper  at  1  cent  each  ? 

14.  WhaJ1-  will  be  the  whole  cost  of  6  spelling  books 
at  8  cents  apiece,  and  of  5  slates  at  10  cents  apiece  ? 

15.  What  will  be  the  difference  of  the  cost  of  6 
yards  of  cloth  at  5  dollars  a  yard,  and  of  4  yards  of 
cloth  at  6  dollars  a  yard  ? 

16.  What  will  be  the  whole  cost  of  ten   sticks  of 
candy  at  2  cents  apiece,  and  of  4  pounds  of  raisins  at 

II  cents  a  pound  ? 

17.  A  farmer    bought   9    sheep    at   three    dollars 
apiece,  and   two   calves   at  4    dollars    apiece:    how 
much   more  did  he  pay  for  the  sheep  than  for  the 
calves? 

18.  A  farmer  bought  4  cows  at  20  dollars  apiece, 
and    12  sheep   at   4    dollars   apiece :    what   did  the 
whole  cost  him  ? 

19.  Fifteen  plus  8  plus  9,  less  5,  are  how  many?  ^ 

20.  Forty  plus  7,  less  6,  plus  8  plus  9,  are  how 
many  ?  '  w  -a 


LES.   V.]  INTELLECTUAL    ARITHMETIC.  39 

21.  Twenty -five  plus  6  plus  8,  less  3,  plus  1,  are 
how  many  ? 

22.  Fifty  plus  6  plus  9,  less  8,  less  3,  less  4,  are 
low  many  1 

23.  Sixty  plus  8  plus  7  plus.l,  less  fi,  are  how 
nany  ? 

24.  Thirty  less  7  less  6,  plus  8,  plus  9?  are  how 
many  ? 

25.  Mary  has  6  rose-bushes,  and  9  buds  on  each  ; 
also  3  geranium^  with  8  buds  on  each :  how  many 
buds  are  there  in  all  ? 

26.  A  family  consumes  12  pounds  of  meat  in  a 
day.     Beef  is  12  cents  a  pound  and  mutton  9  :  how 
much  will  they  save  by  using  mutton  instead  of  beef1? 

27.  James  bought  4  oranges  at  3  cents  apiece,  and 

2  quarts  of  chestnuts  at  9  cents  a  quart :  what  did 
he  pay  in  all  ? 

28.  James  and  John  start  from  the  same  place,  and 
run    in   opposite    directions.     James    runs   30    rods 
and  John  20  :  how  far  are  they  apart  ? 

How  far  would  they  be  apart  if  they  had  run  the 
same  way  ] 

29.  Two  men  start  from  the  same  place  and  walk  in 
opposite  directions  :  the  first  walks  4  hours  and  goes 

3  miles  an  hour,  the  second  the  same  time  and  goes 
but  2  miles  an  hour:  how  far  will  they  be  apart? 

30.  Two  men  start  from  the  same  place  and  walk 
the  same  way  ;  the  first  walks  4  miles  an  hour,  the 
second  3  :  how  far  will  they  be  apart  at  the  end  of 
the  first  hour  ?     How  far  at  the  end  of  the  second  ? 
The  third  ] 

31.  A  drover  bought  3  sheep  at  4  dollars  apiece, 
and  4  lambs  at  2  dollars  apiece :  he  gave  in  payment 
2  calves  at  5  dollars  apiece,   and  the  rest  in  cash. 
EIow  much  money  did  he  pay  1 

32.  A  jeweller  bought  a,  watch  for  55  dollars,  a 


%0  INTELLECTUAL    ARITHMETIC.  [SEC.  Ill 

chain  for  15  dollars,  and  a  seal  for  8  dollars;  and 
then  sold  the  whole  for  80  dollars  :  did  he  make  or 
lose,  and  how  much'? 

33.  Four    b<>ys    bought    a  foot-ball   for  75   cents. 
John  paid  20  cents,  James  33  cents,  and  William   18 
cents  ;  how  much  did  Reuben  pay  1 

34.  A  grocer  bought  a  hogshead  of  sugar  for  05 
dollars,  and  a  hogshead  of  molasses  for  55  dollars; 
he  sold  them  both  for  125:  did  he  make  or  lose,  and 
how  much'? 

35.  A  farmer  bought  4  sheep  for  15  dollars,  and  6 
lambs  for    10  dollars;  he  sold  the  whole  for  30  dol 
lars:  did  he  make  or  lose,  and  how  much? 

36.(  A  tailor  has  a  piece  of  cloth  containing  25 
yards,  and  a  second  piece  containing  35  yards;  he 
cut  8  yards  from  the  first  piece,  and  12  from  the 
second  :  how  many  yards  of  cloth  has  he  left? 

37.  A  merchant  has  a  piece  of  cloth,  for  which   he 
paid  36  dollars  ;   he  wishes  to  sell  it  so  as  to  make  a 
profit  of  12  dollars  :   what  must  he  ask  for  it? 

38.  A  farmer  has  150  sheep  in  three  fields.      In  the 
first  he  has  45,  in  the  second  55  :  how  many  has  he 
in  the  third  ? 

3(«).  There  is  an  orchard  with  ten  full  rows  of  trees 
and  7  trees  in  each  row  ;  besides  which  there  are  3 
broken  rows  with  4  trees  in  each  row  :  how  many 
trees  are  there  in  the  orchard  ?  How  many  more  in 
the  v\  hole  than  in  the  broken  rows? 

40.  An  orchard  contains  20  apple  trees,  10  cherry 
drees,  and  15  plum  trees  :  how  many  more  apple  and 
cherry  trees  than  plum  trees? 

11.  J ames  worked  4  days  in  a  week  for  12  cents  a 
day,  and  William  2  days  for  4  cents  a  day  :  how 
much  more  did  James  earn  than  William? 

42.  A  butcher  bought  16* sheep  of  one  man,  14  of 
another,  and  25  of  a  third  ;  he  then  killed  9,  afterwards 
he  killed  21  :  how  many  had  he  left  ? 


LES.  VI.]  INTELLECTUAL    ARITHMETIC.  *4ll 

43.  A  man  earned  16  dollars  a  month  for  6  months 
and  his  son  Jarnes  7  dollars  a  month  for  the  same 
time :  how  much  more  did  the  father  earn  than  the  son  1 


LESSON    VI 
Factoring  Numbers. 

1.  Four  are  how  many  times  21     What  are  tli8 
factors  of  4  ?* 

2.  Six  are  how  many  times  3  ?     How  many  times 
2  ?     What  are  the  factors  of  6  ? 

3.  Eight  are  how    many  times  4?      How  many 
times  2?     What  are  the  factors  of  8  ? 

4.  Nine  are  how  many  times  3?     What   are   the 
factors  of  9  1 

5.  Ten  are  how  many  times  5  ?     How  many  times 

2  1     What  are  the  factors  of  10  ? 

6.  Twelve  are  how  many  times  6  ?     How  many 
times  2  ?     How  many  times  4  ?     How  many  times 

3  ?     What  are  the  factors  of  12? 

7.  Fourteen  are  how  many  times  7  1     How  many 
times  2?     What  are  the  factors  of  14  1 

8.  Fifteen  are  how  many    times  5  ?     How  many 
times  31     What  are  the  factors  of  15  ? 

9.  Sixteen  are  how  many  times  8  ?     How  many 
times  2?     How  many  times  4?     What  are  the  fac 
tors  of  16? 

10.  Eighteen  are  how  many  times  9  ?     How  many 
times  2  ?     How  many   times  6?     How  many  times 
3  ?     What  are  the  factors  of  18  ? 

*SCGGESTION. — In  each  of  the  following:  questions,  two  numbers 
are  named,  and  it  is  required  to  find  a  third  which  multiplied  by 
the  second,  will  give  a  product  equal  to  the  first  The  second 
and  third  numbers  are  called  FACTORS  of  the  first. 

Let  the  pupil  point  out  the  factors  in  every  example. 


INTELLECTUAL    ARITHMETIC.  [SEC.  IlL 

11.  Twenty  are  how  many  times  2  1     How  many 
times  10?     How  many  tim^s  5'?     How  many  times 
4  }      What  are  the  factors  of  20  ? 

12.  Twenty-two  are  how  many  times  11  ?     How 
many  times  2? 

18.  Twenty -four  are  how  many  times  12?  How 
many  times  2?  How  many  times  8?  How  many 
times  3  ? 

14.  Twenty-six    are    how    many    times  2  ?     How 
many  times  13?     What  are  the  factors  of  20? 

15.  Twenty  seven   are  how  many  timp-s  9  ?     How 
many  times  3  ?     What  are  the  factors  of  27  ? 

1(3.  Twenty  eight  are  how  many  times  14?  How 
many  times  2?  What  are  the  factors  of  28? 

17.  Thirty  are  how  many  times  15?     How  many 
times  2?     How  many  times  10?     How  many  times 
3  ?      Wr  hat  are  the  factors  of  30  ? 

18.  Thirty-three  are  how  many  times   11?     How 
many  times  3? 

19.  Thirty -four   are    how  many    times  2?     How 
many  times  17  ? 

20.  Thirty-six    are   how   many  times   18?     How 
many  times  12?     How  many  times  9?     How  many 
times  0  ?    How  many  times  4  ?     How  many  times  3  ? 
How  many  times  2?     What  are  the  factors  of  30  ? 

21.  Thirty-eight  are  how  many  times   19?     How 
vaany  times  2  ?      \Vhat  are  the  factors  of  38  ] 

22.  Forty  are  how  many  times  20  ?     How  many 
himes  10?     How  many  times  8?     How  many  times 
0  ?     How  many  times  4  ?     How  many  times  2  ? 

23.  Forty-two  are    how   many  times  21  ?      How 
many  times  2?     How  many  times  6  ?     How  many 
times  7  ?     What  are  the  factors  of  42  ? 

24.  Forty-four  are   how   many  times  22  ?     How 
many  times  2?     How  many  times  11  ?     How  many 
times  4  ?     Wrhat  are  the  factors  of  44  ? 


LE8.   I.]  INTELLECTUAL    ARITHMETIC.  43 

25.  Forty-six    are    how   many  times    23 ?      How 
many  times  2?     What  are  the  factors  of  46? 

26.  Forty-eight  are  h<>w  many   times  24?     How 
many  times  16'?     How  many  times  12?     How  many 
times  8?     How  many  times  0?     How  many  times 
4?      Flow   many   times    3?     How    many    times    2"? 
What  are  the  factors  of  48  ? 

27.  Forty-nine   are  how    many  times  7?     What 
are  the  factors  of  49? 

28.  Fifty  are  how  many  times  25  1     How  many 
times    10?     How    many  times    5?     What  are  the 
factors  of  50? 

29.  Fifty-four    are    how    many    times  9  ?      How 
many  times  6?     How   many   times  27!     What  are 
the  tactors  of  54  ? 

30.  Fifty-six  are  how  many  times  7  ?     How  many 
times  8  ?      How  many   times    28  ?      WThat   are    the 
factors  ? 

31.  Sixty-  are  how  many  times  10?     How  many 
times   12?      How    many   times  5?      What  are   the 
factc  >rs  ? 

32.  Sixty-four    are    how    many    times   8  ?      How 
many  times   10?     How  many  times  4?     What  are 
the  factors  ? 

33.  Seventy  are  how  many  times  7  ?     How  many 
times  10?      What  are  the  factors  ] 

34.  Seventy-two  are  how   many  times  8?     How 
many  times  9  ?     What  are  the  factors? 

35.  Eighty-four  are   how   many  times   12?     How 
many  times  7  ?      What  are  the  factors  ? 

36.  Ninety-six  are  how    many   times    12?     How 
many  times  8?     What  are  the  factors'? 


44  INTELLECTUAL    ARITHMETIC.  [SEC.  IV, 

SECTION    FOUETII. 


LESSON    I. 
In  which  Divisors  are  used  from  2  to  5.* 

,1.  How  many  2's  are  there  in  2.     2  is  contained 
in  2,  how  many  times  ? 

2.  Tfow  many  2's  are  there  in  4?     2  in  4,  how 
many  times  ? 

3.  How  many  2's  are  there  in  6  1     2  in  6,  how 
niary  times  ? 

4.  How  many  2's  are  there  in  8?     2  in  8,  how 
many  times  ? 

5.  How  many  2's  are  there  in  10  ?     2  in   10,  how 
many  times  ? 

6.  How  many  2's  are  there  in  12?     2  in  32,  how 
many  times'? 

7.  How  many  3's  are  there  in  3  ?     3  is  contained 
in  3.  how  many  times'? 

8.  How  many  3's  are  there  in  6  ?     3  in  6,  how 
many  times? 

9.  Plow  many  3's  are  there  in  9?     3  in  9,  how 
many  1 

10.  How  many  3's  are  there  in  12  1     3  in  12,  how 
many  1 

1 1.  How  many  3's  are  there  in  15  ?     3  in  15,  how 
many  times? 

*  DIVISION"  is  the  process  of  finding  how  many  times  one  num 
ber  contains  another. 

The  number  to  be  divided  is  called  the  dividend. 

The  number  by  which  we  divide  is  called  the  divisor. 

The  number  expressing  how  many  times  the  dividend  contains 
the  divisor  is  called  the  quotient. 

In  case  there  is  a  number  left,  it  is  called  the  remainder.  Hie 
divisor  and  quotient  &re  factors  of  the  dividend. 


LES.    I.]  INTELLECTUAL    ARITHMETIC.  45 

12.  How  many  3's  are  there  in  18?     3  in  18,  how 
many  times'? 

13.  How  many  4's  are  there  in  4?     4  is  contained 
in  4,  how  many  times  ? 

14.  How  many  4's  are  there  in  8  ?     4  in  8,  how 
many  times'? 

15.  How  many  4's  are  there  in  twelve?     4  in 
12,  how  many  times? 

10.  How  many  4's  are  there  in  16  ?     4  in  1C,  how 
many  times  ? 

17.  How  many  4's  are  there  in  20  ?     4  in  20,  how 
many  times? 

18.  How  many  4's  are  there  in  24  ?     4  in  24,  how 
many  times  ? 

19.  How  many  5's  are  therein  5?     5  is  contain 
ed  in  5,  how  many  times? 

20.  How  many  5's  are  there  in  10  ?     5  in  10,  how 
many  times  ? 

21.  How  many  5's  are  there  in  15  ?     5  in  15,  how 
many  times  ? 

22.  How  many  5's  are  there  in  20  ?     5  in  20,  how 
many  times? 

23.  How  many  5's  are  there  in  25  ?     5  in  25,  how 
many  times  ? 

24.  How  many  5's  are  there  in  30?     5  in  30,  how 
many  times  ? 

QUESTIONS. 

1.  *  William  has  8  apples,  and  divides  them  equally 
between  two  boys  :  how  many  does  he  give  to  each? 

2.  James  has  12  peaches,  and  divides  them  equal Iv 
between  his  two  sisters  :  how  many  does  he  give  to 
each  ? 

*  ANA  LYSIS. — It  will  require  2  apples  to  give  each  boy  1  ; 
therefore,  each  boy  will  get  1  apple  as  often  as  2  apples  is'con- 
tained  times  in  8  apples  ;  2  apples  are  contained  in  8  apples 
4  times — consequently,  each  boy  will  get  4  times  1  apple,  which 
is  4  apples. 


46  INTELLECTUAL    ARITHMETIC.  [SEC.    IV 

3.  Charles  has   a   basket  containing  20  pears,  and 
divides  them  equally  between  his  father  and  mother: 
how  many  does  he  irive  to  each  ? 

4.  A  father  bought  28  fish-hooks,  and  divided  them 
between  John  and  Charles:   how  many  had  each? 

5.  A   mother   has   a  dozen    needles,  and   gives  an 
equal   number  to  Jane  and   Mary  :  how   many   will 
each  have  ? 

(5.  A  lady  having  two  parlors,  bought  24  chairs,  and 
put  an  equal  number  in  each  room:  how  many  were 
there  in  each  room  ? 

7.  There  are  16  boys  in  a  school-room,  and  but  2 
benches  :  how  many  boys  must  sit  on  each  seat? 

8.  *  flow  many  peaches,  at  2  cents  apiece,  can  you 
buy  for  18  cents  1 

0.  At  3  cents  apiece,  how  many  oranges  can  you 
buy  for  9  cents?  flow  many  can  you  buy  for  12 
cvnts?  How  many  can  you  buy  for  30  cents  ?  flow 
many  can  you  buy  for  24  cents? 

10.  A  boy   has    12   cents,  and  finds  that  he  must 
give  3  cents  apiece  for  tops  :  how  many  can  he  buy  ? 
If  he  has  21  cents,  how   many  can   he   buy  ?     ff  he 
has  24  cents,  how  many  can  he  buy  ? 

11.  If  5  barrels  of  flour  cost  30  dollars,  how  much 
will  1  barrel  cost  ?f 

*  ANALYSIS.  —  Since  one  peach  costs  2  cents,  you  can  buy  as  many  peaches 
for  IS  cents  as  2  is  contained  times  in  18:  2  is  contained  in  18,9  times: 
therefore,  you  can  buy  9  peaches  at  2  cents  apiece  for  IS  cents. 

t  Since  «'>  barrels  of  flour  cost  30  dollars,  one  barrel  will  cost  as  many  dol 
lars  a?  ?  is  contained  times  in  30;  5  is  contained  in  :->0,  6  times;  therefor*, 
tf  5  bar:6iS  of  flour  cost  3')  dollars,  one  barrel  will  cost  6  do:  ars. 

NOTE.  —  The  following  rules  result  from  the  analysis  of  «x- 
1,  8  and  11  : 


Ex.  1.  Divide  the  whole  number  of  things  bi/  the  mnnbfr  of 
parix  11  (o  which  they  arc  to  be  divided;  the  quotient  will  be  th* 
number  in  each  part. 

Ex.  8.  Divide  the  entire  cost  by  the  cost  of  a  tingle  thing  ;  and 
th*  quotient  is  the  number  of  things. 

Ex  11.  Divided  the  entire  cost  by  the  number  of  things,  and 
the  quotient  will  be  the  cost  of  a  single  thing. 


LES.    I.]  INTELLECTUAL    ARITHMETIC.  47 

12.  If  4  yards  of  cloth  cost  24  dollars,  how  much 
will   1  yard  cost  ?     How  much  will  2  yards  cost  ? 
3?  4?  6? 

13.  If  3  yards  of  ribbon  cost  36  cents,  how  much 
will  it  cost  a  yard?     How  much  will  11  yards  cost? 

14.  If  4  pounds  of  beef  cost  48  cents,  how  much 
will  1  pound  cost?     2?     3?     5? 

15.  Twenty  dollars  are  paid  for  5  yards  of  cloth, 
how  murh  is  paid  for  2  yards  ?     4  ?     6  ?     10  ? 

16.  Fifteen  dollars  are  paid  for  5  pairs  of  boots  : 
how  much  must  be  paid  for  1  pair? 

17.  If  4  apples  be  equally  divided  between  4  boys, 
how  many  will  each  have  ? 

18.  At  4  cents  apiece,  how  many  oranges  can  you 
buy   for  8  cents?     How  many  can  you  buy  for  16 
cents  ? 

19.  If  it   takes  4  sheets  of  paper  for  a  book,  how 
many  books  will  20  sheets  make  ?     How  many  will 
28  sheets  make?     How  many  will  32  sheets  make? 
How  many  vvill  36  sheets  make?     How  many  will 
40  sheets  make  ? 

20.  There  are  4  benches  in  a  school-room,  and  20 
scholars:  how   many   must   sit  on  e^ach  beraeh  ?     If 
there  be  24  scholars,  how  many  must  sit  on  each  bench? 
If  there  are  32  scholars,  how  many  must  sit  on  each 
bench  ?     If  there  be  36  scholars,  how  many  must  sit 
on  each  bench  ? 

21.  If  John   pays  4  cents  for  one  top,  how  many 
tops  will  he  buy  for  12  cents?     How  many   will  he 
buy  for  16  cents?     How  many   will   he  buy  for  20 
cents?     How  many  will  he  buy  for  28  cents?     How 
many  will  he  buy  for  40  cents  ? 

22.  If  Charles  gives  4  cents  a  quart  for  chestnuts, 
how  many  will   he  buy  for  8  cents?     How  many  for 
16  cents  ?     How  many  for  36  cents  ? 

23.  In  a  school-house  there  are  5  benches  and  20 


INTELLECTUAL    ARITHMETIC.  [SEC.    IV. 

scholars  :  bow  many  must  sit  on  a  bench  ?  If  there 
are  25  scholars,  how  many  must  sit  OK  a  bench?  If 
there  are  30,  how  many  would  there  sit  on  a  bench? 

24.  Jf  cloth  is  5  dollars  a  yard,  how  many  yards 
can  be  purchased  f  >r  10  dollars?     How  much  can  be 
purchased  for  20  dollars  ?     For  30  dollars  ?     For  40 
dollars?     For  50  dollars? 

25.  If  flour  is  5  dollars  a  barrel,  how  many  barrels 
can  be  purchased  for  15 dollars?     How  many  barrels 
can  be  purchased  for  20  dollars  ?     How  many  barrels 
for  30  dollars? 

26.  If  tape  is  5  cents  a  bunch,  how  many  bunches 
can  be  bought  for  20  cents  ?     How  many  bunches 
can  IDC  bought  for  50  cents?     How  many  bunches 
can  be  bought  for  45  cents  ? 

27.  If  5  sheets   of  paper  make  a  copy-book,  how 
many  books  will  20  sheets  make  ?     How  many  books 
will  30  sheets  make  ? 


LESSON    II. 
In  which  Divisors  are  used  from  6  to  10. 

1.  How  many  6's  are  there  in  G  ?     6  is  contained 
in  0,  how  many  times  ? 

2.  How  many  6's  are  there  in  12  \     6  in  12,  how 
many  times  ? 

3.  How  many  6's  in   18?     6   in   18,   how  many 
times  ? 

4.  How  many   6's  in  24  ?     6  in  24,   how   man  f 
times  ? 

5.  How  many   6's  in   30?     6   in   30,  how  many 
times  ? 

0.  How   many  6's   in  36  ?     6   in   36,   how   many 
times  ? 


LES.   II. J  INTELLECTUAL    ARITHMETIC.  41* 

7.  ITow   many  7's    are    there    in    ~  1      7  is  con* 
tained  in  7,   how  many  times'? 

8.  How  many  7's  are  there  in  14  ?     7  in  14,  how 
many  times? 

9.  ITow  many  7's  are  there  in  21  ?     7  in  21,  how 
many  times  ? 

10.  How  many  7's  are  there  in  28  ?     7  in  28,  how 
many  times  ? 

11.  How  many  7's  are  there  in  35  ?     7  in  35,  how 
many  times  ? 

12.  How  many  7's  are  there  in  42?     7  in  42,  how 
many  times  ? 

13.  How  many  8's  are  there   in   8  ?     8    is    con 
tained  in  8,  how  many  times? 

14.  How  many  8's  are  there  in  16?     8  in  16,  how 
many  times? 

15.  How  many  8's  are  there  in  24  ?     8  in  24,  how 
many  times? 

16.  How  many  8's  are  there  in  32  ?     8  in  32,  how 
many  times  ? 

17.  How  many  8's  are  there  in  40  ?     8  in  40,  how 
many  times  ? 

18.  How  many  8's  are  there  in  48  ?     8  in  48,  how 
many  times? 

19.  How  many  9's  are  there  in  9  ?     9  in  9,  how 
many  times  ? 

20.  How  many  9's  are  there  in  18  ?     9  in  IS,  how- 
many  times? 

21.  How  many  9's  are  there  in  27  ?     9  in  27,  how 
many  times  ? 

22.  How  many  9's  are  there  in  36  ?     9  in  36,  how 
many  times? 

*     23.  How  many  9's  are  there  in  45  ?     9  in  45,  how 
many  times  ? 

24.  How  many  9's  are  there  in  54?     9  in  54,  how 
many  times? 

5 


50  INTELLECTUAL    ARITHMETIC.  [SEC     IV* 

25.  How  many  10's  are  there  in  30?     10  in  30> 
how  many  times  ? 

26.  How  many  10's  are  there  in  60?     10  in  60, 
how  many  times  1 

27.  How  many  10's  are  there  in  100?     10  in  J00> 
how  many  times  ? 

QUESTIONS. 

1.  If  6  sheets   of  paper  make  a   copy-book,   how 
many  books  will  12  sheets  make  ?     How  many  books    , 
will   24   sheets   make?     How   many    books  "will   30 
sheets    make  ?      How    many    books   will    48    sheets 
make?      How  many  books  will  60  sheets  make? 

2.  If  1    yard   of   broadcloth  costs  6   dollars,    how 
many   yards  can    be   bought   for  30  dollars?     How 
many   yards  can    be  bought  for  36   dollars?     II ow 
many  yards  for  42   dollars?     How  many  yards  for 
54  dollars?     How  many  for  60  dollars? 

3.  If  a  man  travels  6  miles  in  1  hour,  how  many 
hours   will   it  take   him    to  travel    12  miles?      How 
many   hours   will   it  take    him    to   travel  24  miles? 
How  long  will  it  take  him  to  travel  ->0  miles?      How 
long  to  travel  54   miles  ?     How   long  to  travel  60 
miles  ? 

4.  Forty -two  apples  are  divided  equally  among  6 
boys  :  how  many  does  each  one  receive  ? 

5.  If  54  peaches  be  divided  equally  between  6  boys, 
how  many  will  each  receive? 

6.  If  a  yard  of  ribbon  costs  6  cents,  ho.w  many 
yards  can  be  bought  for  24  cents  ?     How  many  for 
30  cents? 

7.  If  you  have  28  dollars,  how  many  yards  of  cloth 
can  you  buy  at  7  dollars  a  yard  ?     Plow  many  yards 
for  35  dollars?     How  many  for  70  dollars? 

8.  If  you  have  16  apples  to  divide  among  8  boys, 
how  many  do  you  give  to  each  ?     If  you  have  32, 


LES.  TII.J  INTELLECTUAL    ARITHMETIC. 

how   many  ?     If  you  have  64,  how  many  ?     If  you 
have  96,  how  many  ? 

9.  A  laborer  engaged   to  work   for   12   dollars  a 
month  ;  at  the  end  of  the  time  he  received  96  dollars  , 
how  many  months  did  he  work  I 

10.  James  engaged  to  work  for  9  cent?  a  day,  and 
at  the  end  of  the  time  received  72  cents :  how  many 
days  did  he  work  ? 

11.  If  a  man  can  do  a  piece  of  work  in  56  hours, 
how  many  days  will  it  take  him  to  do  it,  working  7 
hours  a  day  ? 

12.  If  two  boys  are  36  yards  apart,  and  the  om* 
behind  gains  on  the  other  4  yard  a  minute,  how  many 
minutes  before  he  will  overtake  him  ? 

13.  A  man  has  84   pounds  of  butter  to  be  put  ir 
seven  jars  :  how  much  must  be  put  in  each  jar  ? 

14.  A  lady  paid  108  cents  for  9  yards  of  ribbon: 
how  much  did  she  pay  a  yard  ? 

15.  If  a  milliner  pays  121  cents  for  11  yards  of 
ribbon  :   how  much  does  she  pay  for  1  yard  1 

16.  How  many  dresses  can  be  cut  from  132  yards 
of  silk,  if  each  dress  contains  12  yards  ? 


LESSON    III. 
Addition,  Subtraction,  Multiplication,  Division. 

1.  Two  in  5,  how  many  times,  and  what  over  ? 

2.  Two  in  7,  how  many  times,  and  what  over  ?* 
8.  Four  in  15,  how  many  times? 

4.  Five  in  19,  how  many  times? 

5  Five  in  36,  how  many  times  ? 

6.  Seven  in  42,  how  many  times? 

7.  Nine  in  60  how  many  times? 

*  When  there  is  a  remainder,  let  it  be  simply  mention**! 


INTELLECTUAL    ARITHMETIC.  [SEC.  I7i 

8.  Seven  in  64,  how  many  times  ? 

9.  Ten  in  f>5,  how  many  limes? 

10.  Six  in  70,  how  many  times? 

11.  Nine  in  IOC,  how  many  times? 
12    Five  in  56,  how  many  times  ? 

13.  Twelve  are  how  many  times  2? 

14.  Eighteen  are  how  many  times  6  ? 

15.  Nineteen  plus  6,  are  how  many  times  5  ? 

16.  Twenty  plus  8,  are  how  many  times  7  ?     How 
many  times  4? 

17.  Thirty  less  6,  are  how  many  times  6?     How 
many  times  4?     How  many  times  3  1 

18.  Sixty  less  5,  are  how  many  times  11  ? 

19.  Ninety  plus  9,  are  how  many  times  11  ? 

20.  Eighty-seven  plus  3,  are  how  many  times  101 

21.  Forty -five  plus  4,  are  how  many  times  7? 

22.  Sixty-nine  plus  15,  are  how  many  times   12  ! 
How  many  times  6?     How  many  times  7? 

23.  Six  times  7  less  2,  are  how  many  times  10? 

24.  Forty  plus  4  times  6,  are  how  many  times  81 

25.  Fifty  plus  3  times  4,  plus  2,  are  how  many 
times  8? 

26.  Five  times  6  plus  4  times  9,  are  how  many 
times  1? 

27.  Seven  times  8  plus  4,  are  how  many  times  6? 
12?     10?     5? 

28.  Eight  times  5  plus  5,  are  how  many  times  9  ? 
5? 

29.  Five  times  11  less  5,  are  how  many  times  2! 
10?     5? 

30.  Forty-six  less  3  times  2,  arc  how  many  times 
10?     8?     5?     4?     2? 

31.  Seven  times  9  plus  3  times  4,  are  how  many 
times  25  ? 

32.  Six  times  4  less  3  times  4,  are  now  many 
times  3  ?     4  ?     6  ? 


LES.  III.]  INTELLECTUAL    ARITHMETIC. 

QUESTIONS. 

1.  If  2  yards  of  cloth  cost  6  dollars,  what  will  8 
yards  cost?* 

2.  If  3  oranges  cost  12  cents,  what  will  11  cost? 

3.  If  4  boxes  of  raisins  can  be  bought  for  16  dollars, 
IKW  much  will  9  boxes  cost? 

4.  It"  7  pounds  of  sugar  cost  56  cents,  what  will  12 
pounds  cost  ? 

5.  What  will  6  lemons  cost  if  8  cost  24  cents? 

6.  James  has  apples  worth  2  cents  apiece,  how  many 
must  he  give  for  6  oranges  worth  3  cents  apiece? 

7.  How  many    eggs  at  8  cents  a  dozen  must  be 
given  for  12  pounds  of  sugar  worth  6  cents  a  pound  ? 

8.  How  many  knives  at  2  shillings  apiece  are  worth 
4  axes  at  8  shillings  apiece  ? 

9.  How  much  barley  at  5  shillings  a  bushel   must 
be  given  for  6  bushels  of  wheat  at  10    shillings   a 
bushel  ? 

10.  A  farmer  bought  4  yards  of  cloth  at  3  dollars 
a  yard  and  paid  in   labor  at  2  dollars  a  day  ;  how 
many  days  must  he  labor? 

1 1.  John  had  oranges  worth  4  cents  a  piece  which 
he  gave  to  James  lor  2  quarts  of  cherries  worth  8 
cents  a  quart ;  how  many  oranges  did  he  give  for  the 
cherries  ] 

NOTE. — If  any  number  be  divided  into  two  equal  parts,  one 
of  the  parts  is  called  one  half.  If  it  be  divided  into  three  equal 
parts,  one  of  the  parts  is  called  one  third,  and  two  of  the  parts 
are  called  two  third*.  If  it  be  divided  into  four  equal  parts. 
one  of  the  parts  is  called  one  fourth,  two  of  the  parts,  two  fourths, 
three  of  them,  three  fourths,  <fcc.,  <fec. 

*  ANALYSIS. — One  yard  will  cost  one  half  as  much  as  tw« 
yards.  If  two  yards  cost  6  dollars,  on^  yard  will  cost  one  half 
of  six  dollars,  which  are  3  dollars;  8  yards  will  cost  8  times  as 
much  as  1  yard :  if  1  yard  costs  3  dollars,  8  yards  will  cost  8 
times  3  dollars  which  are  24  dollars:  therefore,  if  2  yards  of 
eloth  cot>t  6  dollars,  8  yards  will  cost  24  dollars. 
5* 


INTELLECTUAL    ARITHMETIC.  [SEC.    IV. 

12.  How  many  yards  of  cotton  cloth,  at  8  cents  a 
yard,  must  be  given  for  6  pounds  of  butter  worth  12 
cents  a  pound  1  •    > 

13.  A  farmer  bought  4  yards  of  broadcloth  worth 
5  dollars  a  yard,  and  paid  for  them  in  calves  worth  4 
dollars  apiece.     How  many  must  he  give  1 

14.  How  many  barrels  of  flour  worth  6  dollars  a 
barrel  must  be  given  for  8  barrels  of  fish  worth  3 
dollars  a  barrel  ? 

15.  A  man  bought  18  oranges  at  the  rate  of  o  for 
5  cents.     How  much  did  they  come  to  ?* 

16.  How  many  pears  at  the  rate  of  2  for  3  cents 
can  be  bought  for  24  cents  ? 

17.  James  bought  20  marbles  at  the  rate  of  5  for  6 
cents.     How  much  did  they  cost? 

18.  How  much  cloth  worth  2  dollars  a  yard,  must 
be  given  for  2  firkins  of  butter  worth  18  dollars  a 
firkin  ? 

19.  What  will  8  pounds  of  beef  cost,  if  6  pounds 
cost  54  cents  1 

20.  A  farmer  has  6  dozen  of  eggs  worth  12  cents 
a  dozen,  and  wishes  to  exchange  them  for  nutmegs 
at  3  cents  apiece  :  how  many  shall  he  receive  ? 

21.  A  man  bought  4  barrels  of  flour  at  7  dollars  a 
barrel,  and  wishes  to  exchange  them  for  cloth  at  2 
dollars  a  yard :  how  many  yards  of  cloth  should  he 
receive  ? 

22.  Four  men  bought  a  boat  for  60  dollars  :  the^ 
paid.  12  dollars  for  repairing  her,  and  then  sold  her  so 
that  they  made  5  dollars:  what  did  they  get  for  her? 
How  much  more  than  they  gave  ? 

*  ANALYSIS. — Since  3  oranges  cost  5  cents,  18  oranges  will 
cost  as  many  times  5  cents  as  3  is  contained  times  in  18,  ivhick 
is  6  times :  therefore,  18  oranges  will  cost  6  times  5  cents  which 
ore  30  cents. 


LES.  III.]  INTELLECTUAL    ARITHMETIC. 

23.  A  laborer  engaged  to  work  for  a  year  at  144 
dollars  a  month,  but  was  sent  away  at  the   end  of  7 
months  :  how  much  should  he  receive  ?     How  much 
less  than  if  he  had  staid  the  entire  year  ? 

24.  A  drover  has  8  calves   for  which  he  paid  40 
dollars,  and  9  sheep  for  which  he  paid  27  dollars  • 
#hat  did  he  pay  apiece  for  the  calves  (\     What  for 
2he  sheep  ?     How  much  for  all  ? 

25.  ]f  8  firkins  of  butter  are  worth  96  dollars,  how 
many  firkins  must  be  given  for  3  barrels  of  sugar 
worth  20  dollars  a  barrel  ? 

26.  Six  men  agree  to  do  a  piece  of  work  for  90 
dollars  :  it  turns  out  that  each  man  makes  5  dollars 
by  the   bargain :    what   was   the    cost  of  doing    the 
work  ? 

27.  A  man  has  40  eggs,  from  which  he  wishes  to 
rsyse  chickens :  his  eggs  are  worth  a  cent  apiece,  and 
his  chickens,  when  hatched,    will  be  worth  3  cents 
apiece.     Now,  if  15  of  his  eggs  prove  addle,  how 
much  will  he  gain? 

28.  How  much  honey  at   16  cents  a  pound  must 
be  given  for  6  pounds  of  coffee  at  8  cents  a  pound  \ 

29.  How  many  hats  af  4  dollars  apiece,  can  a  man 
buy  if  he  gives  in   payment   seven  handkerchiefs  at 
1  dollar  apiece,  and  5  pair  of  boots  at  6  dollars  a  pair  ? 
What  will  he  have  left? 

30.  A  grocer  purchased  4  barrels  of  sugar  at  12 
dollars  a  barrel,  and  5  hogsheads  of  molasses  at  20 
dollars    a    hogshead,    and    sold    the    whole    for    150 
dollars  :  did  he  make  or  lose,  and  how  much  ? 

31.  Five  men  agree  to  do  a  piece  of  work   for  60 
dollars,  each  to  receive  an  equal   part.     When   the 
work  is  half  done,  two  of  the  men  quit,  and  the  other 
three  fini>h  it :  how  much  should  each  receive  ? 

32.  Three  men  made  up  a  purse.     The  first  put  hv 
8  dollars ;  the  second  twice  as  much  as  the  first,  and 


INTELLECTUAL    ARITHMETIC.  [SEC.  I?. 

the  third  twice  as  much  as  the  second  :  how  much 
\va.s  put  in  in  all  ]  What  would  each  have  put  in  had 
they  contributed  equally  ? 

33.  Mr.  Johnson  bought  14  yards  of  broadcloth  at 
4  dollars  a  ya>d,  and  having  cut  off  9  yards,  sold  the 
remainder  for  one  half  of  what  he  paid  for  the  whole 
less  3  dollars  :  how  much  more  was  this  for  each 
yard  than  he  gave  1 

34  If  0  men  can  do  a  piece  of  work  in  9  days,  how 
long  will  it  take  one  man  to  do  it  ?* 

35.  How  many  men  in  9  days  can  perform  as 
much  labor  as  12  men  can  in  6  days? 

30.  If  a  barrel  of  flour  will  last  5  men  25  days, 
how  long  will  it  last  9  men  ] 

37.  A  man  has  a  piece  of  work  which  would 
employ  3  men  for  8  days:  how  many  men  can  do  it 
in  1  -Jay  ?  In  4  days?} 

38  If  8  men  can  do  a  piece  of  work  in  5  days,  how 
long  will  it  take  one  man  to  do  it?  How  many  men 
will  do  it  in  one  day  ] 

39.  A  piece  of  work  requires  9  men  for   10  days, 
how  many  men  can  do  the  same  work  in  one   day  ? 
In  3  days]     In  6  da.vs? 

40.  if  6  meti  can   build  a  wall   12  rods  long  in  8 
days,  how   long   will    it   take  one  man   to   buiid   it? 
How   many  men  can  build  it  in  1  day  ?     In  3  days! 
In  12  days? 

*  ANALYSIS. — It  will  take  1  man  6  times  as  long  to  do  th$ 
same  work  as  it  will  6  men:  if  it  takes  6  men  9  days,  it  will 
take  1  man  6  times  9  days,  which  are  54  days  :  therefoie,  if  it 
take  6  men  9  days  to' do  a  piece  of  work,  it  will  take  1  man  54 
days  to  do  the  same  work. 

*f  ANALYSIS. — It  will  take  8  times  as  many  men  to  do  tbt 
work  in  1  day  as  it  will  to  do  it  in  8  days:  if  it  takes  3  m»  n  8 
days,  it  will  take  8  times  3  men,  which  are  24  men,  to  do  the 
work  in  one  day:  therefore,  if  8  meu  can  do  a  piece  of  work  in 
8  da  vs.  24  men  can  do  the  same  work  in  1  day. 


LKS.  IV.]  INTELLECTUAL    ARITHMETIC. 


:*  LESSON    IV. 

Combinations. 

The  classes  are  to  be  dri'Jled  in  the  following  tables 
until  tfiey  are  fully  understood.  The  tables  are  to  ^e 
read  across  the  page:  thus.  1  and  1  are  two  ;  1  from  2 
leaves  one;  once  1  is  one  ;  1  in  1,  once  ;  and  similarly 
for  the 


1   and 

1  ; 

1   fr 

om 

0 

;  once     1 

j 

1 

in 

1 

1   and 

2  ;   1    from 

3 

;  once     2 

; 

1 

in 

2 

1   and 

3; 

1    fr 

om 

4 

;  once     3 

i 

1 

in 

3 

1   and 

4; 

1   fr 

orn 

5 

;  once     4 

1 

in 

4 

1    and 

5  ; 

1    fr 

orn 

6 

;  once     5 

j 

1 

in 

5 

1   and 

6  ; 

1    fr 

om 

7 

;  once     6 

1 

in 

6 

1   and 

7; 

1    fr 

oni 

8 

;  once     7 

j 

1 

in 

7 

1   and 

8; 

1    from 

9 

;  once     8 

? 

1 

in 

8 

1   and 

9  ;   1   from 

10 

;  once     9 

1 

in 

9 

1   and 

10; 

1    from 

11 

;  once   10 

j 

1 

in 

10 

2 

and      1 

;   2 

from 

3 

> 

two  times 

1 

j 

'2 

in 

2 

2 

and     2 

;  2 

from 

4 

i 

two  times 

2 

^ 

2 

in 

4 

2 

and     3 

;  2 

from 

5 

? 

Uvo  times. 

3 

; 

2 

in 

0 

2 

and     4 

;  2 

from 

6 

'•> 

two  tim^s 

4 

) 

'2 

in 

8 

2 

and     5 

;  2 

from 

7 

two  times 

5 

j 

2 

in 

iO 

2 

and     6 

;   2 

from 

8 

; 

two  times 

6 

•2 

in 

S2 

2 

and     7 

;  2 

from 

9 

• 

two  times 

7 

'2 

in 

11 

2 

and     8 

;  2 

from 

10 

j 

.  t 
two  times 

8 

; 

2 

in 

2 

and     9 

5      "^ 

from 

ti 

'J- 

two  times 

9 

J 

2 

in 

18 

2  and   10  ;  2  from    12  ;  two  times   10  ;  2  in  20 

*  SuGSESTiON.—Tbe  object  in  placing  these  tablel^sre  is  tc 
drill  tlie  pupil  in  changing  his  mind  from  one  subject  to  another 
In  each  line  he  begins  with  addition,  passes  to  subtraction,  then 
(^multiplication,  and  then  to  division.  The  difficulty  of  doing 
this  ac&erdtety,  and  rapidly,  will  illustrate  fully  the  value  of  the 
exercise.  A  mind  must  be  well  disciplined  that  can  go  through 
tho  "ntire  tables  without  a  mistake. 


INTELLECTUAL    ARITHMETIC.  [SEC.  IF. 


3 

and 

i; 

3 

from 

4  ; 

8 

times 

1 

;  3 

in 

3 

3 

and 

:j 

from 

*^  5 

a 

times 

2 

;  3 

in 

6 

3  and 

si 

:; 

from 

o  ; 

3 

times 

3 

;  3 

in 

9 

3 

and 

4; 

a 

from 

7  ; 

3 

times 

4 

;  3 

in 

12 

3 

and 

:j 

from 

8  ; 

3 

times 

5 

;  3 

in 

15 

3 

ancl 

G  ; 

:; 

from 

3 

times 

G 

;  3 

iii 

18 

3 

and 

7; 

;J 

from 

10  i 

3 

times 

7 

;  3 

in 

21 

3 

and 

8; 

:j 

from 

11  ; 

3 

times 

8 

;  3 

in 

24 

3 

and 

9; 

a 

from 

12; 

a 

times 

9 

;  3 

in 

27 

3 

and 

10; 

;> 

from 

13  ; 

a 

times 

10 

;  3 

in 

30 

'4 

and 

i; 

4 

from 

5  ; 

4 

times 

1 

;  4 

in 

4 

4 

and 

2  ; 

4 

from 

C  ; 

4 

times 

2 

;  4 

in 

8 

4 

and 

3; 

4 

from 

7  ; 

4 

times 

3 

;  4 

iii 

12 

4 

and 

4; 

4 

from 

8  ; 

4 

times 

4 

;   4 

in 

16 

4 

and 

5; 

4 

from 

9  ; 

4 

times 

5 

:   4 

in 

'20 

4 

and 

4 

from 

10  ; 

4 

times 

G 

;  4 

in 

24 

4 

and 

7; 

4 

from 

11  ; 

4 

times 

7 

;  4 

in 

28 

4 

and 

8; 

4 

from 

12  ; 

4 

times 

8 

;  4 

in 

32 

4 

and 

9  ; 

4 

from 

13  ; 

4 

times 

0 

;  4 

in 

36 

4 

and 

10; 

4 

from 

14  ; 

4 

times 

10 

;  4 

in 

40 

5 

and 

*       j 

5. 

from 

G   ; 

5 

times 

1 

;  5 

in 

5 

5 

and 

^  ; 

5 

from 

7   ; 

5 

times 

2 

;  5 

in 

10 

5 

and 

3  ; 

5 

from 

8  ; 

9 

times 

3 

;   5 

in 

15 

5 

and 

4  ; 

5 

from 

9   ; 

f> 

times 

4 

;  5 

in 

20 

5 

and 

5  ; 

5 

from 

10   ; 

f) 

times 

5 

;   5 

in 

25 

5 

and 

.  6  ; 

5 

from 

11  ; 

fi 

times 

G 

;  5 

in 

30 

5 

and 

'     ; 

5 

from 

12   ; 

a 

times 

7 

;  5 

in 

35 

5 

and 

8  ; 

5 

from 

13  ; 

5 

times 

8 

;  5 

in 

40 

5 

and 

9    ; 

5 

from 

14   ; 

5 

times 

.9 

;   5 

in 

45 

5 

and 

10   ; 

5 

from 

15   ; 

5 

times 

10 

;  s 

in 

50 

6 

and 

1  ; 

G 

from 

7  ; 

(> 

times 

1 

;  6 

in 

6 

G 

and 

2  ; 

G 

from 

8   ; 

0 

times 

0 

;   G 

in 

12 

G 

and 

3  ; 

G 

from 

"      5 

6 

times 

o 

;  c 

in 

18 

G 

and 

4  ; 

6 

from 

10    ; 

6 

times 

4 

;    G 

in 

24 

6 

and 

«*     J 

G 

from 

Jl    ; 

G 

times 

5 

;  ^ 

in 

30 

LES.  IV.1  INTELLECTUAL    ARITHMETIC  59 

J  -    ** 


G 

and 

6  ; 

6 

from 

12  ; 

6 

times 

6 

,  6 

in 

36 

G 

and 

'      ? 

8 

from 

13   ; 

6 

tit  nes 

7 

;  6 

in 

42 

6 

and 

8  ; 

G 

from 

14  ; 

6 

times 

8 

5  ^* 

in 

48 

6 

and 

9  ; 

() 

from 

15    ; 

6 

times 

9 

;  6 

in 

54 

6 

and 

10  ; 

G 

from 

15   ; 

6 

times 

16 

;  6 

in 

60 

;7 

and 

i  ; 

7 

from 

8  ; 

7 

times 

1 

;  7 

in 

7 

7 

and 

2  ; 

/ 

from 

9  ; 

7 

times 

2 

;  7 

in 

14 

7 

and 

3  ; 

7 

from 

10  ; 

7 

times 

3 

>    • 

in 

21 

7 

and 

!  ; 

7 

from 

1  1    ; 

7 

times 

4 

.  ^ 

in 

28 

7 

and 

7 

from 

12  ; 

7 

times 

5 

'•>    ' 

in 

35 

7 

and 

6  • 

7 

from 

13   ; 

7 

times 

6 

>    ' 

in 

42 

7 

and 

7 

from 

14  ; 

7 

times 

7 

;  7 

in 

49 

7 

and 

8  !, 

7 

from 

15   ; 

7 

times 

8 

i    * 

in 

56 

7 

and 

9  ; 

7 

from 

16  ; 

7 

times 

9 

;  7 

in 

63 

ry 
t 

and 

10  ; 

7 

from 

**      5 

7 

times 

10 

*  7 

in 

70 

8 

and 

i  ; 

8 

from 

9  ; 

8 

times 

1 

;  8 

in 

8 

8 

and 

2  ; 

8 

from 

10   ; 

8 

times 

2 

;  8 

in 

16 

8 

and 

3  ; 

8 

from 

11  ; 

8 

times 

3 

;  8 

in 

24 

8 

and 

4  ; 

8 

from 

12  ; 

8 

times 

4 

;  8 

in 

32 

8 

and 

5  ; 

8 

from 

13   ; 

8 

times 

5 

;  8 

in 

40 

8 

and 

o  ; 

8 

from 

14   ; 

8 

times 

6 

;  8 

in 

48 

8 

and 

'      j 

8 

from 

15   ; 

8 

times 

7 

>    ~ 

in 

56 

8 

and 

8  ; 

8 

from 

16  ; 

8 

times 

8 

;  8 

in 

64 

8 

and 

9   ; 

8 

from 

17   ; 

8 

times 

9 

;  8 

in 

72 

8 

and 

10  ; 

8 

from 

18   ; 

8 

times 

10 

;  8 

in 

80 

9 

and 

i  ; 

9 

from 

10    ; 

9 

times 

1 

;  9 

in 

9 

9 

and 

2  ; 

1) 

from 

11  ; 

9 

times 

2 

;  9 

in 

18 

p 

and 

3  ; 

9 

from 

12   ; 

i) 

times 

3 

in 

27 

0 

and 

4  ; 

9 

from 

13    ; 

9 

times 

4 

;  9 

'ni 

36 

'•> 

and 

5  ; 

9 

from 

14   ; 

9 

times 

5 

;  9 

in 

45 

9 

and 

6  ; 

9 

from 

15   ; 

9 

times 

6 

;  9 

in 

54 

9 

and 

'      > 

9 

from 

16   ; 

9 

times 

7 

;  9 

in 

63 

9 

and 

8  ; 

9 

from 

17   ; 

9 

times 

8 

;  9 

in 

73 

9 

and 

9  ; 

9 

from 

18   ; 

0 

times 

9 

;  9 

in 

81 

9 

and 

10    ; 

9 

from 

19  ; 

1) 

times 

10 

;  9 

in 

90 

60  INTELLECTUAL    ARITHMETIC.  [SEC.  IV. 


10 

and 

ii 

10 

from 

11  ; 

10 

times 

i; 

10 

in 

10 

10 

and 

2; 

10 

from 

12; 

10 

times 

2  ; 

10 

in 

20 

10 

and 

3; 

10 

from 

13; 

10 

times 

3; 

10 

in 

30 

10 

and 

4; 

10 

from 

14; 

10 

times 

4; 

10 

in 

40 

10 

and 

10 

from 

15; 

10 

times 

5; 

10 

in 

50 

10 

and 

6  ; 

10 

from 

16; 

10 

times 

10 

in 

60 

10 

and 

7; 

10 

from 

17  ; 

10 

times 

1\ 

10 

in 

70 

10 

and 

8; 

10 

from 

18; 

10 

times 

8; 

10 

in 

80 

10 

and 

9  ;' 

10 

from 

19; 

10 

•times 

10 

in 

90 

10 

and 

10; 

10 

from 

20; 

10 

times 

10  | 

10 

in 

100 

11 

and 

1  ; 

11 

from 

12; 

11 

times 

1; 

11 

in 

11 

11 

and 

2  ; 

11 

from 

13; 

11 

times 

2  ; 

11 

in 

22 

Ii 

and 

3; 

11 

from 

14; 

11 

times 

3; 

11 

in 

33 

11 

and 

4; 

11 

from 

15; 

11 

times 

4; 

11 

in 

44 

11 

and 

5; 

11 

from 

16; 

11 

times 

5; 

11 

ii: 

55 

11 

and 

6; 

11 

from 

17  ; 

11 

times 

6; 

11 

in 

66 

11 

and 

7; 

11 

from 

18; 

11 

times 

7; 

11 

hi 

77 

11 

and 

8; 

11 

from 

19; 

11 

times 

8; 

11 

\r, 

88 

11 

and 

11 

from 

20; 

11 

times 

11 

in 

99 

11 

and 

10; 

11 

from 

21; 

11 

times 

10; 

11 

ii; 

no 

11 

and 

11  ; 

11 

from 

22; 

11 

times 

11  ; 

11 

in 

121 

11 

and 

12; 

11 

from 

23; 

11 

times 

12; 

11 

in 

132 

12 

and 

i; 

12 

from 

13; 

12 

times 

i  ; 

12 

in 

12 

12 

and 

2; 

12 

from 

14; 

12 

times 

2; 

12 

in 

24 

12 

and 

3; 

12 

from 

15; 

12 

times 

3; 

12 

in 

36 

12 

and 

4; 

12 

from 

16; 

12 

times 

4; 

12 

in 

48 

12 

and 

5; 

12 

from 

17; 

12 

times 

5; 

12 

in 

60 

12 

and 

6  ; 

12 

from 

18; 

12 

times 

6; 

12 

in 

72 

12 

and 

7; 

12 

from 

19; 

12 

times 

7; 

12 

in 

84 

12 

and 

8; 

12 

from 

20; 

12 

times 

8; 

12 

in 

96 

12 

and 

12 

from 

21; 

12 

ti  m  es 

9  ; 

12 

in 

108 

12 

and 

10; 

12 

from 

22; 

12 

times 

10; 

12 

in 

12( 

12 

arid 

12 

from 

23; 

12 

times 

11  ; 

12 

in 

132 

12  and  12;  12  from  24-   12  times  12     12  in  144 


LSS.    I.]  INTELLECTUAL    ARITHMETIC. 

SECTION    FIFTH. 


UNITED    STATES    MONEY. 

10  mills  make  ...     1  cent    .     .     .     ct. 

10  cents 1  dime  T>;  .     .      d. 

10  climes      ....     1  dollar       .     .     $. 
10  dollars     ....     1  eagle  .     .     .     E. 

1.  How    many    mills    are   there   in    2  cents?     In 
§  cents1?     In  half  a  cent?     In  5  cents? 

2.  How  many  cents  are  there  in  ten  mills  ?     In  15 
mills?     In  65  mills? 

3.  How  many  cents  are  there  in  5  dimes  ?     In  6 
dimes  ?'    In  8  dimes  ?     In  10  dimes  ? 

4.  flow  many  dimes  are  there  in  10  cents  ?     In  12 
cents  ?     In  16  cents  ?     In  20  cents? 

5.  How  many  dimes  in   1  dollar?     In  2  dollars? 
In  3  dollars  ?     In  4  dollars  ?     In  5  dollars  ? 

6.  How  many  dollars  in   1   eagle?     In  2  eagles? 
In  5  eagles  ?     In  6  eagles'?     In  9  eagles  ? 


ENGLISH    STERLING    MONEY. 

4  farthings  make  .     .  1  penny  .  .  d. 

12  pence 1  shilling  .  .  s. 

20  shillings    ....  1  pound  .  .  £. 

21  shillings    ....  1  guinea  .  . 

1.  How  many  farthings  are  there  in  1  penny  ?     In 
£  pence  ?     In  4  ?     In  6  ?     In  8  ? 

2.  How  many  pence  in  4  farthings  ?   In  8  farthings  ? 
In  12?     In  14?     In  16? 

3.  How  many  pence  are  there  in  1  shilling?     In  2 
shillings?     In  3  shillings?     in  4  shillings? 

4.  How  many  shillings  are  there  in  12  pence?     la 
18?     In  20?     In  24?     In  26  pence  ? 


INTELLECTUAL    ARITHMETIC.  [SEC. 


TABLE    OF    LONG    MEASURE. 


12  inches,     in.     make  1  foot,  .     ...    ft. 

3  feet 1  yard,-.  .  .  .  yd. 

5^  yards,  or  16|  feet  .  1  rod,  .  .  .  .  rd. 

40  rods 1  furlong,  .  .  .  fur. 

8  furlongs,  or  320  rods,  1  mile,.     .     .     .     m. 

3  miles 1  league,    .     .     .    L. 

60  geographical,  or  69i  )  ,    -,  7  0 

2  Y  1  decree,    .   deq.  or  °. 
statute  miles    .     .  j 

360  degrees      ....     a  great  circle. 

1.  How  many  feet  in  24  inches  ?  In  six  yards  how 
many?  How  many  yards  in  15  feet  ?  In.  24?  How 
many  rods  in  2  furlongs?  How  many  furlongs  in 
160  rods  ?  How  many  miles  in  9  leagues  ?  How 
many  leagues  in  54  miles  ? 


TABLE    OF    CLOTH    MEASURE. 

2j  inches,     in.     make  »1  nail,     ....     na. 

4    nails    .     .     .     ^'  .  1  quarter  of  a  yard,   qr. 

4  quarters  ....  1  yard,    .     .     .           yd. 
3    quarters  ....  1  Ell  Flemish,      E.  FL 

5  quarters  .     .     .     .  1  Ell  English,        E.  E. 

1.  How    many    inches    make   2   nails?     3    nails? 
4  nails  ? 

2.  How    many    nails    make  a  quarter  of  a  yard  ? 
How  many  make  2  quarters?     How  many  make  3 
quarters  ? 

3.  How  many  quarters  make  2  yards  ?     3  yards  ? 
4  yards  ?     5  yards  ?      . 

4.  How  many  quarters   make   an    Ell    Flemish  ? 
How  many  make  2  Ells  ?    3  Ells  ?    4  ?    5  ?    6  ?    7  ? 

5.  How  many  Ells  Flemish  in   3    quarters  of  a 
yard  *     In  6  quarters  ?     In  9  ?     In  12  ?     In  15  ? 


LES.    I.]  INTELLECTUAL    ARITHMETIC.  63 


TABLE    OF    SQUARE    MEASURE. 

144  square  inches,  sq.  in.  make  1  square  foot,    sq.  ft. 

9  square  feet 1  square  yard,  sq.  yd. 

30^-  square  yards  ....  1  square  rod,  .  P. 
40  square  rods "  1  rood,  .  .  j& 

4  roods 1  acre,  .  .  .  A. 

640  acres 1  square  mile,  .  M. 

1.  How  many  square  inches  in  2  square  feet? 

2.  How  many  square  feet  in  2  square  yards.     In 
3?     In  4?     In  5?     In  9 1     In  10? 

3.  How   many    square    rods    in    2  roods  ?     How 
many  in  3  roods  ?     In  5  ? 

4.  How  many  roods  in  80  square  rods?     In  120? 
In  160,  how  many  ? 


TABLE    OF    CUBIC    MEASURE. 

1728  cubic  inches, cu.in.  make  1  cubic  foot,  .  cu.ft. 

27  cubic  feet      ....  1  cubic  yard,  .  cu.  yd. 

40  feet  of  round,  or         )  1  ^ 

50  feet  of  hewn  timber   f           OD'     '     '  *  ' 

42  cubic  feet      .     . —     .  1  ton  of  shipping,       T. 

10  cubic  feet       ....  1  cord  foot,  .  c.  ft. 

8  cord  feet,  or  ) 

128  cubic  feet        }       '     '  '  C0rd'  '     '  '  '      G 

1.  In  1  cubic  foot,  how  many  cubic  inches  ?     How 
many  in  2 1     In  3  ? 

2.  In  one  cubic  yard,  how  many  cubic  feet  1     How 
many  in  2?     In  4?     In  6? 

3.  How  many  cord  feet  in  3  cords  of  wood  ?    In  5  ? 
In  6? 

4.  How  many  cubic  feet  in  2  cords  ?     In  half  a 
cord,  how   many?     How    many  in    a   quarter  of  a 
cord? 


64  INTELLECTUAL    ARITHMETIC.  [SEC.  V. 


TABLE    OF    WINE    MEASURE. 

4  gills,       gi.       make  1  pint,      .     .  .  pi. 

2  pints 1  quart,    ...  qt. 

4  quarts 1  gallon,  .  .  .  gal. 

311  gallons  ....  1  barrel,.  .  .  bar. 

63  gallons  ....  1  hogshead,  .  .  hhd. 

2  hogsheads       ...  1  pipe,     .     .  .  pi. 

2  pipes,  or  4  hogsheads  1  tun,       .     .  .  tun. 

1.  How    many    gills  make   a  pint?     How  many 
make  2  pints'?     3?     4?     5?     6] 

2.  How  many  pints  in  a  quart  1     In  2  quarts  ?     In 
3?     In  4?     In  6?     In  11 

3.  How  many  quarts  in  two  pints  ?     In  4  ?     In  6? 
In  8?     In  10?     In  12? 

4.  How  many  quarts  in  1  gallon]     In  2?     In  3  ? 
In  41     In  5?     In  7  ? 

5.  How  many  gallons  in  4  quarts?     In  8  ?     In  12? 
In  16?     In  20? 

6.  Plow  many  gallons  in  a  barrel  ? 

7.  How  many  gallons  in  a  hogshead?     In  2? 


TABLE    OF    ALE    AND    BEER    MEASURE, 

2  pints,    pt.    make  1  quart,     .     .     .     .     qt. 

4  quarts  .  .  .  .  1  gallon,  .  .  .  .  gal. 

36  gallons  ...  1  barrel,  ....  bar. 

54  gallons  ...  1  hogshead,  .  .  .  hhd. 

1.  How  many  pints  in  2  quarts?  In  3  ?     In  4  ? 
In  5? 

2.  How  many  quarts  in  2  pints  ?  In  4  ?     In  6  ? 
In  8?     In  10?     In  12? 

3.  How  many  quarts  in  2  gallons  ?  In  4  ?     In  5  ? 

4.  How  many  gallons  in  4  quarts?  In  8?    In  12? 


LES.  I.J  INTELLECTUAL    ARITHMETIC.  65 


TABLE    OF    DRY    MEASURE. 

2  pints,    pi.     make     1   quart,  ....      qt. 

8  quarts    ....     1   peck,    ...»  pk. 

4  pecks     ....     1   bushel,       .     •     .  bu. 

36  bushels  ....     1   chaldron,  /JjH  v-";:  ch. 

1.  How  many  pints  in  2  quarts?     In  3?  In  4  ? 
In  5  ?     In  6 1 

2.  How  many  quarts  in  4  pints?  In  6?  In  8?  In  10? 


TABLE    OF    AVOIRDUPOIS    WEIGHT. 

16  drams,  dr.  make  1  ounce,.  .  .  .  oz. 
16  ounces  ....  1  pound,  Ib. 

25  pounds    ....     1   quarter,    .     .     .     qr. 
4  quarters  ....      1   hundred  weight,    cwt. 
20  hundred  weight      .     1  ton,      ....      T. 

1.  How  many  drams  in  an  ounce?     How  many- 
ounces  in  a  pound  ?    How  many  pounds  in  a  quarter  ? 
How  many  quarters  in  a  hundred  ?     How  many  hun 
dred  in  a  ton  ? 

2.  How  many  drams  in  2  ounces  ?     How  many 
ounces  in  2  pounds?     How  many  pounds  in  2  quar 
ters?      How    many  quarters    in  2  hundred?'    Hovr 
many  hundreds  in  2  tons  ? 

TABLE    OF    TROY    WEIGHT. 

24  grains,  gr.  make  1  pennyweight,  .  pwt. 
20  pennyweights  .  1  ounce,  .  <  M  oz, 
12  ounces  ....  1  pound,  ...  Ib. 

1.  How  many  grains  are  there  in  1  pennyweight? 
In  2  pennyweights?     In  3  penny  weights  ?     In  4  pen 
ny  weights  ? 

2.  How  many  pennyweights  are  there  in  24  grains  ? 
In  48  grains  ?     In  72  grains  ?     In  96  grains  ? 

6* 


66  INTELLECTUAL    ARITHMETIC.  [SEC.  V. 

3.  How  many    pennyweights  in    1   ounce?     In  2 
ounces]     In  3  1*    In  4/    In  5? 

4.  How    many    ounces    are    there    in    20   penny 
weights  ?     In  40  pennyweights  ?     In  00  1     In  80  pen 
nyweights  ? 

5.  How  many  ounces  are  there  in   1  pound  ?     In 
2  pounds  ?    In  3  pounds  ?  In  4  pounds  ?  In  5  pounds  ? 

0.  liow  many  pounds  in  12  ounces?  In  24  ?  In 
36  ]  In  48  1  In  60  ? 

7.  How  many  pennyweights  in  4  ounces  ?  In  2 
ounces  1  How  many  grains  in  1  ounce  ?  In  2 
ounces?  In  3  ounces? 


TABLE  OF  APOTHECARIES  WEIGHT. 

20  grains,     gr.     make     1   scruple,  ...      9. 

3  scruples    ....      1   dram.      ...      3 . 

8  drams 1   ounce,     ...       §  • 

12  ounces       ....     1   pound,    ...      Ik. 

1.  How  many  grains  are  there  in  1  scruple?     In 
2  scruples?     In  3  ?      In  4? 

2.  How  many  scruples  in  20  grains  ?     In  40  ?     In 
60  ?     In  80  ? 

3.  How  many  scruples  in  1  dram?     In  2  ?     In  3  ? 
In  4,  how  many  ? 


.  .  m. 

.  .  hr. 

.  .  da. 

.  .  u<L 

.  mo. 

.  -  yr. 

.  .  0. 

The    year   is   also    divided    into    twelve   calendar 
months,  which  contain  an  unequal  number  of  days. 


60  seconds, 
60  minutes 
24  hours 
7  davs 

TABLE    C 

sec.  make 

>F    TIME. 

1   minute, 
1   hour,    . 
1   day,   .  . 
1    week,  . 
1   month, 
1   year,  . 
1   century 

4  weeks 
52  weeks 
100  years 

.     .     .     . 

LES.  I.] 

Winter, 
Spring, 

Summer, 

Autumn, 
Winter, 


INTELLECTUAL    ARITHMETIC. 

(     1     month 


67 


January, 

.     31 

February,  . 

March,    .     .     . 

.     31 

April,     .     .     . 

.     30 

May,      .     .     . 

.     31 

June, 

.     30 

July,       .     .     . 

.     31 

August,  .     .     . 

.     31 

September,  .    . 

.     30 

October, 

.     31 

November, 

.     30 

December, 

.     31 

Total,  365 


Thirty  days  hath  September, 

April,  June,  and  November; 

All  the  rest  have  thirty-one, 

Excepting  February,  twenty-eight  alone. 


TABLE    OF    CIRCULAR    MEASURE. 


60  seconds,  "    make 

60  minutes 

30  degrees  .... 

12  signs,  or  360°  .     . 


1  minute,  marked  '. 

1   degree,   .     .     .     .  °. 

1   sign, s. 

1  circle,                      .  c. 


1.  How  many  seconds  in  3  minutes  ?    In  4  1    In  5  ? 

2.  How  many  minutes  in  6  decrees  ?    In  4  ?    In  5  ? 

3.  How  many  degrees  in  4  signs?     In  6  ?     In  7? 
In8? 

4.  How  many  degrees  in  240  minutes  ?     In  720  ? 
How  many  signs  in  90°  ?     In  150°  ?     In  180  ?° 


*  February  has  29  days  in  leap-year. 


INTELLECTUAL    ARITHMETIC.  [SEC.  VL 

SECTION    SIXTH. 

LESSON   I. 
Of  the  Fraction  One  Half. 

1.  If  an  apple  be  divided  into  two  equal  parts,  one 
of  the  parts  is  called  one  half.     What  are  the  2  parts 
called? 

2.  How  many  halves  are  there  in  one  apple? 

3.  If  a  pear  be  divided  into  two  equal  parts,  what 
*e  tire  parts  called? 

4.  How  many  halves  are  there  in  a  pear  ? 

5.  How  many  halves  are  there  in  1  ?     In  1  and  a 
naif,  how  many  ? 

6.  How  many  halves  are  there  in  2  pears  ?     In  2 
and  a  half,  how  many  ?* 

7.  How  many  halves  are  there  in  3  things  ?     In  3 
and  a  half,  how  many  ? 

8.  How  many  halves  are  there  in  4?     In  4  and  a 
half,  how  many  ? 

9.  How  many  halves  are  there  in  5?     In  5  and  a 
half,  how  many  ? 

10.  How  many  halves  are  there  in  6  ?     In  6  and  a 
half,  how  many  ? 

11.  How  many  halves  are  there  in  7?     In  7  and  a 
half,  how  many  ? 

12.  How  many  halves  in  8  ?     In  8  and  a  half? 

13.  How  many  halves  in  9  ?     In  9  and  a  half? 

14.  How  many  halves  in  10?     In  10  and  a  half? 

15.  How  many  halves  in  11   and   a  half?     In  12 
and  a  half? 

16.  How  many  whole  things  are  there  in  2  halves  f 
In  4  halves?!     In  6  halves?     In  12  halves?     In  18 
halves?     In  24?     In  30?     In  28  ?     In  17  ?     In  9? 
In  13?     In  15? 

*  ANALYSIS. — There  are  2  halves  in  1 ;  therefore,  in  any  nuua 
ber  of  things  there  are  twice  as  many  halves  as  whole  things 
In  2  and  a  half  there  are  6  halves. 

\  ANALYSIS. — Since  there  is  1  whole  thing  in  two  halves,  the 
number  of  whole  things  are  found  by  dividing  the  halves  by  2. 


LKB.  II.]  INTELLECTUAL    ARITHMETIC.  69 

QUESTIONS. 

1.  If  one  half  an  orange  cost  1  cent,  how  much  will 
3  halves  cost  ? 

2.  If  one  half  a  pine-apple  cost  4  cents,  how  much 
will  1  pine-apple  cost  1 

3.  If  one  half  a  yard  of  cloth  cost  2  dollars,  how 
fliuch  will  6  yards  cost  1     4  yards  ?     5  yards  1 

4.  If  one  and  one  half  yards  of  cloth  cost  6  dollars, 
what  will  5  yards  cost?* 

5.  James  gave  3  and  a  half  cents  for  a  top,  G  and 
a  half  cents  for  paper,  9  half  cents  for  a  pencil,  and  7 
aalf  cents  for  an  orange :  how  much  did  he  pay  in  all  ? 

6.  Arm  bought  a  skein  of  silk  for  9  half  cents,  a 
jpool  of  thread  for  7  and  a  half  cents,  a  thimble  for 
J5  half  cents  :  how  much  did  she  pay  in  all  1 


LESSON    II. 
Of  the  Fraction  One  Third. 

1.  If  an  apple  be  divided  into  three  equal  parts, 
one  of  the  parts  is  called  one  third  :  what  are  2  of 
the  parts  called  ?     3  of  them  ? 

2.  How  many  thirds  are  there  in  one  apple  ? 

3.  How  many  thirds  are  there  in  1  ? 

4.  How  many  thirds  are  there  in  2  1     In  2  and 
one  third  ?     Why  ? 

5.  How  many  thirds  are  there  in  3 1     In  3  and  two 
thirds  ?     In  3  and  one  third  ?     Why  ? 

*  ANALYSTS. — One  and  one  half  yards  are  equal  to  8  half 
yards.  One  half  yard  will  cost  one  third  as  much  as  3  half 
yards.  If  3  half  yards  cost  6  dollars,  one  half  yard  will  cost  one 
third  of  6  dollars,  which  are  2  dollars.  If  one  half  a  yard  costs 
2  dollars,  one  yard  will  cost  2  times  2  dollars,  which  are  4  dol 
lars  ;  and  if  1  yard  costs  4  dollars,  5  yards  will  cost  6  times  4 
dollars,  which  are  20  dollars. 

3 


70 


INTELLECTUAL    ARITHMETIC.  [SEC.  VI. 


6.  How  many  thirds  are  there  in  4 1     In  4  and  one 
third  1     In  4  and  two  thirds  1 

7.  How  many  thirds  are  there  in  5  ?     In  5  and  two 
thirds?     In  5  and  one  third  ? 

8.  How  many  thirds  are  there  in  6  ?     In  6  and  one 
third  ?     In  6  and  two  thirds  ? 

9.  How  many  thirds  are  there  in  7  ?     In  7  and  one 
third  ?     In  7  and  two  thirds  ? 

10.  How  many  thirds  are  there  in  8  ?     In  8   and 
one  third  ?     In  8  and  two  thirds  ? 

11.  How  many  thirds  are  there  in  9  ?     In  9  and 
one  third  ?     In  9  and  two  thirds  ? 

12.  How  many  thirds  are  there  in  10?     In  10  and 
two  thirds  ? 

13.  How  many  thirds  are  there  in  11 1     In  11  and 
Cwo  thirds? 

14.  How  many  thirds  are  there  in  12?     In  12  and 
one  third?     In  12  and  two  thirds? 

15.  How  many  whole  things  are  there  in  6  thirds? 
In  9  thirds?     In   15   thirds?     In    18  thirds?     In  8 
thirds?     In  7  thirds? 

16.  How  many  whole  things  in  14  thirds?     In  16 
thirds  ?     In  25  thirds  ?     In  17  thirds  ?     In  19  thirds? 

17.  How  many  thirds  in  9  and  6  thirds?     In  7 and 
5  thirds  ?     In  6  and  1 1  thirds  ? 

18.  How  many   thirds  in  6  thirds  and  5  thirds  1 
In  9  thirds,  and  4  and  2  thirds,  how  many  ?     In  It! 
thirds  and  9  thirds  and  5  thirds,  how  many  thirds? 

QUESTIONS. 

1.  If  one  third  of  an  orange  cost  one  cent,  what 
will  the  whole  orange  cost?     What  will  two  oranges 
>ost  ? 

2.  If  one  third  of  a  yard  of  cloth  cost  two  dollars, 
what  will  three  yards  cost? 

3.  If  one  third  of  a  barrel  of  flour  cost  two  dollars, 
vhat  will  six  barrels  cost  2 


LES.  III.]  INTELLECTUAL    ARITHMETIC.  71 

4.  If  two  thirds  of  a  pound  of  tea  cost  40  cents, 
what  will  2  pounds  cost  1 

5.  If  3  and  two  thirds  pounds  of  sugar  cost  33  cents, 
how  much  will  8  pounds  cost? 

6.  If  3  and  one  third  pounds  of  coffee  cost  40  cents, 
how  much  will  9  pounds  cost  ? 

7.  If  8  and  two  thirds  yards  of  cloth  cost  52  dollars, 
ho"<v  much  will  7  yards  cost  ? 

8.  John  gave  two  and  two  thirds  dimes  for  a  pen 
knife:  how  much  did  he  pay  for  3  penknives? 

9.  If  2  sheep   cost  5  and  one  third  dollars,  what 
will  6  cost  1 

10.  If  1  pound  of  honey  cost  12  and  two  third 
cents',  what  will  9  pounds  cost  ? 


LESSON   III. 
Of  the  Fraction  One  Fourth. 

1.  If  an  apple  be  divided   into  four  equal   parts, 
one  of  the  parts  is  called  one  fourth.     What  arc  2  of 
them  called  1     3  of  them  1     4  of  them  ? 

2.  How  many  fourths  are  there  in  1 1 

'  3.  How   many  fourths    are    there    in    1    and    one 
fourth  ? 

4.  How  many  fourths  are  there  in  2?     In  2  and 
one  fourth?     In  2  and  three  fourths  ? 

5.  How  many  fourths  are  there  in  3  ?     In  3  and 
two  fourths  ?     In  3  and  three  fourths? 

6.  How  many  fourths  are  there  in  4  ?     In  4  and 
one  fourth  ?     In  4  and  two  fourths? 

7.  How  many  fourths  are  there  in  5?     In  5  and 
one  fourth?     In  5  and  three  fourths? 

8.  How  many  fourths  are  there  in  G  ?     In  6  and 
one  fourth?     In  G  and  three  fourths? 

9.  How  many  fourths  are  there  in  7?     In   7  and 
one  fourth.     In  7  and  three  fourths? 


72  INTELLECTUAL    ARITHMETIC.  [SEC      ,1. 

10.  How  many  fourths  are  there  in  8 1    In  8  arid  one 
fourth  ?    In  8  and  two  fourths  1    In  8  and  three  fourths  ? 

11.  How  many  fourths  are  there  in  9?     In  9  and 
one  fourth  ?     In  9  and  two  fourths  ?     In  9  and  three 
fourths  ? 

12.  How  many  fourths  are  there  in  10?     In    10 
and  one  fourth  1     In  10  and  two  fourths  ?     In  10  and 
three  fourths  ? 

13.  How  many  whole  things  in  4  fourths  ?     In  8 
fourths?     In    20    fourths?     In   17   fourths?     In    19 
fourths  ? 

14.  How  many  whole  things  in  12  fourths  ?     In  25 
fourths?     In    30    fourths?     In   35    fourths?     In  48 
fourths  ?     In  60  fourths  ?     In  56  fourths  ? 

QUESTIONS. 

1.  If  one  fourth  of  a  yard  of  cloth  costs  2  dollars, 
how  much  will  1  yard  cost  ?     How  much  will  4  yards 
cost? 

2.  If  3  fourths  of  a  pound  of  coffee  cost  12  cents, 
what  will  5  pounds  cost? 

3.  If  7  fourths  yards  of  cloth  cost  14  dollars,  what 
will  9  yards  cost? 

4.  If  3  yards  of  cloth  cost  3  and  3  fourth  dollars, 
what  will  5  yards  cost? 

5.  How  many  fourths  are  there  in  6  and  one  fourth  ? 

6.  How  many  whole  things  are  there  in  2  and  5 
fourths,  and  5  and  9  fourths  ? 

7.  If  a  barrel  of  vinegar  cost  4  dollars,  what  will  5 
and  3  fourths  barrels  cost  ? 

8.  If  9  fourths  yards  of  muslin  cost  18  cents,  what 
will  12  yards  cost? 

9.  If  1 1  fourths  yards  of  broadcloth  cost  22  dollars 
what  will  7  yards  cost  ? 

10.  If  3  and  a  fourth  yards  of  cambric  cost  39  cents, 
how  much  will  7  yards  cost  ? 


LEB.  IV.J  INTELLECTUAL    ARITHMETIC.  78 

LESSON     IV. 
Of  the  Fraction  One  Fifth. 

1.  If  an  apple  be  divided  into  5  equal  parts,  one  of 
the  parts  is  called  one  fifth.     What  are  2  of  then, 
called?     3  of  them?     4  of  them?     5  of  them? 

2.  How  many  fifths  are  there  in  1  apple  ?     Hew 
many  fifths  in  1  thing  ? 

3.  How  many  fifths  are  there  in  2?     In  2  and  one 
fifth  ?     In  2  and  2  fifths  ? 

4.  How  many  fifths  are  there  in  3  ?     In  3  and  one 
fifth  ?     In  3  and  two  fifths  ?     In  3  and  three  fifths  ? 

5.  How  many  fifths  are  there  in  4  ?     In  4  and  one 
fifth  ?     In  4  and  two  fifths?     In  4  and  three  fifths? 

6.  How   many  fifths  are  there  in   5  ?     In  5  and 
three  fifths  ?     In  5  and   one  fifth  ?     In  5  and  four 
fifths  ? 

7.  How  many  fifths  are  there  in  6  ?     In  6  and  one 
fifth  ?     In  6  and  two  fifths  ?     In  6  and  four  fifths  ? 

8.  How  many  fifths  are  there  in  7  ?     In  7  and  one 
fifth  ?     In  7  and   two  fifths  ?      In  7  and  three  fifths  ? 
In  7  and  four  fifths  ? 

9.  How  many  fifths  are  there  in  8  ?     In  8  and  two 
fifths  ?     In  8  and  three  fifths  ?     In  8  and  one  fifth  ? 

10.  How  many  fifths  are  there  in'  9  ?     In  9  and 
one  fifth?     In  9  and  three  fifths?      In  9  and   four 
fifths  ? 

11.  How  many  fifths  are  there  in  10?     In  10  and 
one  fifth?     In  10  and  three  fifths?     In  10  and  four 
fifths  ? 

12.  How  many  whole  apples  are  equal  to  5  fifths 
.  if  an  apple  ?     How  many  whole  things  in  5  fifths  ? 

13.  How  many  whole  things  in   10  fifths]     In  12 
ftfths?     hi  14  fifths?     In  11  fifths? 

14.  How  many  whole  things  in  15  fifths?     In  18 
fifths?     In  25  fifths?     In  37   fifths?     In  40  fifths? 
In  60  fifths  ? 

7 


74  INTELLECTUAL    ARITHMETIC.  [bEC.    VI 

15.  How  many  fifths  in  8  and  4  fifths'?     In  6  and 
3  fifths  ?     In  4  and  2  fifths  1     In  1 1  and  4  fifths ) 

QUESTIONS. 

1 .  If  1  fifth  of  a  yard  of  broadcloth  cost  1  dollar, 
what  will  1  yard    cost] 

2.  If  3  fifths  of  a  pound  of  coffee  cost  15  cents, 
\\hat  will  3  pounds  cost? 

3.  If  4  fifths  of  a  pound  of  tea  cost  40  cents,  what 
will  4  pounds  cost  ? 

4.  If  1  yard  of  cloth  cost  3  fifths  of  a  dollar,  what 
will  6  yards  cost  ? 

5.  If  1  pound  of  tea  cost  3  fifths  of  a  dollar,  what 
will  5  pounds  cost? 

0.  If  2  yards  of  cloth  cost  4  fifths  of  a  dollar,  what 
will  10  yards  cost? 

7.  John  has  9  fifths  apples,  James  G  fifths,  Charles 
7  fifths,  and  Reuben  3  fifths  :  how  many  entire  apples 
have  they  in  all  ? 

8.  If  my  pencil  cost  4  fifths  of  a  dime,  my  pen  1 
fifth,  my  paper  3  fifths,  and  wafers  2  fifths :    how 
much  do  I  pay  in  all  ? 

9.  If  a  family  consume  3  fifths  of  a  barrel  of  flour 
in  1  week,  how  much  will  they  consume  in  5  weeks? 

10.  Eight   times    3   fifths   are    how    many   iifths  ? 
How  many  times  1  ? 


LESSON    V. 
Of  the  Fraction  One  Sixth. 

1.  If  an  apple  be  divided  into  six  equal  parts,  one  of 
the  parts  is  called  one  sixth:  what  are  2  of  them  called  ? 
3  of  them  ?     4  of  them  ?     5  of  them  ?     6  of  them  ? 

2.  How  many  sixth  parts  are  there  in  1  apple  ? 

3.  How  many  sixths  are  there  in  1  ? 

4.  How  many  sixths  are  there  in  1  and  one  sixth? 


LES.  V.]  INTELLECTUAL    ARITHMETIC.  75 

5.  How  many  sixths  are  there  in  2  1     In  2  and  3 
sixths  ?     In  2  and  4  sixths  1 

6.  How  many  sixths  are  there  in  3  1     In  3  and  4 
sixths?     In  3  and  5  sixths'?     In  3  and  2  sixths? 

7.  How  many  sixths  are  there  in  4  ?     In  4  and  3 
sixths  ?     In  4  and  5  sixths  ?     In  4  and  2  sixths  ? 

8.  How  many -sixths  are  there  in  5?     In  5  and  3 
sixths  ?     In  5  and  4  sixths  ?     In  5  and  5  sixths? 

9.  How  many  sixths  are  there  in  6?     In  6  and  3 
sixths  ?     In  0  and  5  sixths  ? 

10.  How  many  sixths  are  there  in  7  ?     In  7  and  1 
sixth  ?     In  7  and  2  sixths  ?     In  7  and  5  sixths  ? 

11.  How  many  sixths  are  there  in  8  ?     In  8  and  1 
sixth  ?     In  8  and  2  sixths  ? 

12.  How  many  sixths  are  there  in  9  ?     In  9  and  2 
sixths  ?     In  9  and  4  sixths  ? 

13.  How  many  sixths  are  there  in  10?     In  10  and 
1  sixth?     In  10  and  2  sixths?     In  10  and  3  sixths? 
In  10  and  4  sixths? 

14.  How  many  whole  apples  are  there  in  6  sixths 
apples  ?     In  12  sixths  ?     In  1 8  sixths  ?     In  24  sixths  1 
In  36  sixths  ?     In  72  sixths  ? 

15.  How  many  whole  things  in  15  sixths?     In  54 
sixths  ?     In  66  sixths  ?     In  45  ?     In  50  ?     In  48  ?     lu 
63?     In  75? 

QUESTIONS. 

1.  If  a  family  consume  1  bushel  and  3  sixths  of  a 
bushel  of  potatoes  in    1    week,  how  much  will  they 
consume  in  8  weeks? 

2.  How  many  sixths  of  apples  can  you  cut  from 
apples  and  2  sixths?     How  many  from  6  and  5 

sixths? 

3.  If  James  gives  5  sixths  of  a  shilling  for  a  pencil, 
and  4  sixths  of  a  shilling  for  a  slate,  and  3  sixths  of  a 
shilling  for  paper:  how  much  will  he  pay  in,  all? 


76  INTELLECTUAL    ARITHMETIC.  [SEC.    VL 

4.  If  2  sixths  of  a  barrel  of  flour  will  last  a  family 
4  weeks,  how  long  will  2  barrels  last  them  7 

5.  If  4  sixths  of  a  load  of  hay  is  worth  8  dollars, 
what    are  4   loads  worth,  at  the  same  rate  7 

6.  What  is  the  cost  of  3  barrels  of  flour  at  4  dol 
lars  and  2  sixths  a  barrel  7 

7.  A  man  gave  to  each  of  12  men  5   sixths  of  a 
loaf  of  bread  :  how  many  loaves  did  he  distribute? 

8.  A  man  bought  6  barrels  of  flour  at  5   and  5 
sixths  dollars  a  barrel :  what  did  it  come  to  ? 

9.  What  is  the  product  of  5  and  4  sixths,  multiplied 
by  9? 

10.  What  is  the  product  of  6  and  5  sixths,  multi 
plied  by  12  ? 


LESSON    VI. 
Of  the  Fraction  One  Seventh  and  One  Eighth. 

1.  If  an  apple  be  divided  into  7  equal  parts,  one  of 
the  parts  is  called  one  seventh  :  what  are  2  of  the 
parts  called  7     3  of  them  1     4  of  them  1     5  of  them  7 

0  of  them  1     7  of  them  7 

How  many  sevenths  are  there  in  1   apple?     How 
many  sevenths  in  1  thing  ? 

2.  How  many  sevenths  are  there  in  2  7     In  2  and 

1  seventh,  how  many  ?     In  2  and  2  sevenths,  how 
many  1 

3.  How  many  sevenths  are  there  in  3  7     In  3  and 

1  seventh,  how  many  7     In  3  and  2  sevenths,  how 
nany  7 

4.  How  many  sevenths  are  there  in  4?     In  4  and 
:J  sevenths,  how  many  7     In  4  and  4  sevenths,  how 
many  7     In  4  and  6  sevenths  7 

5.  How  many  sevenths  are  there  in  5?     In  5  and 

2  sevenths,  how  many  7     In  5  and  3  sevenths,  how 
many  7     In  5  and  6  sevenths  7 


LE8.  VI.]  INTELLECTUAL    ARITHMETIC.  77 

6.  How  many  sevenths  are  there  in  6  ?    "In  6  and 

2  sevenths,  how  many'?     In  6  and  3  sevenths?     In 
6  and  4  sevenths? 

7.  How  many  sevenths  are  there  in  7  ?     In  7  and 
1    seventh?     In   7    and    3    sevenths?     In  7   and  4 
sevenths? 

8.  How  many  sevenths  are  there  in  8  ?     In  8  and 
1  seventh?     In  8  and  two  sevenths?     In    8  and  5 
sevenths  ? 

9.  How  many  sevenths  are  there  in  9  ?     In  9  and 
4  sevenths  ?     In  9  and  3  sevenths  ? 

10.  How  many  sevenths  are  in  10?     In  10  and   1 
seventh?     In    10    and    2    sevenths?     In    10    and  3 
sevenths? 

11.  How  many  whole  things  in  14  sevenths?     In 
28  sevenths  ?     In  35  sevenths  ?     In  49  sevenths? 

12.  How  many  whole  things  in  17  sevenths?     In 
14    sevenths   and    4    sevenths?     How  many   in   03 
sevenths  ? 

13.  How  many  sevenths  in  9  and  5  sevenths  ? 

14.  How   many  whole  things  in  30  sevenths,  14 
sevenths  and  5  sevenths  ? 

EIGHTHS. 

1.  If  an  apple  be  divided   into  eight  equal  parts, 
what  is  one  of  the  parts  called  ?     How  many  eighths 
are   there    in    one  ?     What    are    two    parts    called  ? 

3  parts ?     4  parts?     5  parts  ?     7  parts  ? 

2.  How  many  eighths  are  there  in  2  ?     How  many 
in  2  and  1  eighth?     In  2  and  5  eighths?     In  2  and  0 
eighths? 

3.  How  many  eighths  are  there  in  3  ?     In  3  and  2 
eighths  ?     In  3  and  4  eighths?     In  3  and  7  eighths  1 
In.  3  and  6  eighths? 

4.  How  many  eighths  in  4  ?     In  4  and  1    eighth  ? 
In  4  and  3  eighths  1     In  4  and  5  eighths  ?     In  4  and 
0  eighths  ? 

7* 


78 


INTELLECTUAL    ARITHMETIC.  [SEC.  VI. 


5.  How  many  eighths  are  there  in  5  1     In  5  and  1 
eighth1?     In  5  and  2  eighths?     In  5  and  3   eighths? 
In  5  and  5  eighths? 

6.  How  many  eighths  are  there  in  G  ?     In  G  and  2 
eighths  ?     In  G  and  3  eighths?     In  6  and  7  eighths? 
Ill  G  and  4  eighths  ? 

7.  How  many  eighths  are  there  in  7  '?     In  7  and  1 
eighth  ?     In  7  and  2-  eighths  ?     In  7  and  3  eighths  ? 

8.  How  many  eighths  are  there  in  8  ?     In  8  and  1 
eighth  ?     In  eight  and  3  eighths  ?     In  8  and  4  eighths? 
In  8  and  G  eighths  ? 

0.  How  many  eighths  are  there  in  0  ?  In  9  and  1 
eighth  ?  In  9  and  2  eighths  ?  In  9  and  3  eighths  2 
In  9  and  4  eighths  ? 

10.  How  many  eighths  are  there  in  10?     In   10 
and  1  eighth?     In  10  and  2  eighths?     In  10  and  3 
eighths? 

11.  How   many     whole    things    are    there   in    1G 
eighths  ?     In  24  eighths,  how  many  ?     How  many  in 
32  eighths  ?     In  40  ?     In  50  ?     In  60  ?     In  70  ?     In 
5G  ?     In  9G  ?     In  88  ?     In  2G  ? 


LESSON  VII. 
Of  the  Fractions  One  Ninth  and  One  Tenth. 

1.  If  an  apple  be  divided  into  9  equal  parts,  one  of 
the  parts  is  called  one  ninth.     How  many  ninths  are 
there  in  1?     What  aie  2  of  the  parts  called?     5  of 
them  ?     7  of  them  ? 

2.  How  many  ninths  are  there  in  2  things  ?     How 
many  in  2  and  2  ninths  ?     In  2  and  4  ninths  ?     In  2 
und  5  ninths  ?     In  2  and  G  ninths  ? 

3.  How  many  ninths  are  there  in  3?     In  3  and 
7  ninths?     In  3  and  G  ninths?     In  3  and  5  ninths'? 
lii  3  and  8  ninths  ? 


LES.  VII.']  INTELLECTUAL    ARITHMETIC.  79 

4.  How  many  ninths  are  there  in  4  things  ?     In 
4  and    1   ninth  1      In  4  and  3  ninths'?      In    4   and 
6  ninths  1     In  4  and  8  ninths  ? 

5.  How  many  ninths  are  there  in  5  ?     In  5  and 
2  ninths?     In  5  and  3  ninths?     In  5  and  4  ninths? 
In  5  and  6  ninths? 

6.  How  many   ninths  are  there  in  6  ?     In  6  and 
4  ninths  ?     In  6  and  5  ninths?     In  6  and  8  ninths  ? 

7.  How    many  ninths  are  there  in  7  ?     In  7  and 
£  ninths?      In   7  and  4  ninths?      In  7  and  6  ninths  ? 
In  7  and  8  ninths? 

8.  How  many  ninths  are  there  in  eight  ?     In  8  and 
1  ninth  ?     In  8  and  2  ninths  ?     In  8  and  4  ninths  ? 

9.  How  many  ninths   are  there  in  9  ?     In  9  and 
8  ninths?     In  9  and  4  ninths  ?     In  9  and  5  ninths? 

10.  How  many  ninths  are  there  in  10?     In  10  and 
1  ninth?     In  10  and  2  ninths?     In  10  and  8  ninths? 

11.  How  many  whole  things  in  18  ninths?     In  27? 
in  36  ?     In  63  ?      In  72  ?      In  90  ?      In  81  ?     In  54 1 
In  67?     In  59?     In  71?     In  99? 

TENTHS. 

1.  If  an  apple  be  divided  into  ten  equal  parts,  one 
of  the  parts  is  called  one  tenth.     How  many  tenths 
are  there  in  one  thing?     What  are  2  parts  called? 
4  ?     5  ?     7  ? 

2.  How  many  tenths  are  there  in  2  ?     In  2  and 

3  tenths  ?     In  2  and  5  tenths  ?     In  2  and  9  tenths  ? 

3.  How  many  tenths  are  there  in  3  ?     In    3  and 

4  tenths?     In  3  and  5  tenths  ?     In  3  and  6  tenths  ? 

4.  How  many  tenths   are  there  in  4?     In  4  and 
4  tenths  ?     In  4  and  5  tenths  ?     In  4  and  8  tenths  ? 

5.  How  many  temjs  are  there  in  5?     In  5  and 
3  tenths  ?  In  5  and  6  tenths  ?     In  5  and  9  tenths  ? 

6.  How  many  tenths  are  there  in  6  ?     In  6  and 
3  tenths  ?  In  6  and  7  tenths  ?     In  6  and  8  tenths  ? 


80  INTELLECTUAL    ARITHMETIC.  [SEC.   VI. 

7.  How  many   tenths  are  there  in  7  1     In  7  and 
1}  tenths  ?     In  7* and  8  tenths'?     In  7  and  9  tenths  ? 

8.  How  many  tenths  are  there   in  8?     In  8  and 
4  tenths  1     In  8  and  5  tenths  ?     In  8  and  9  tenths  ? 

9.  How   many  tenths  are   there   in   9  1     In  9  and 

4  tenths  ?     In  9  and  5  tenths  ?     In  9  and  8  tenths'! 

10.  How  many  tenths  are  there  in  10  ?     In  10  and 

5  tenths  ?     In  10  and  0  tenths  ?     In  10  and  9  tenths  1 

11.  How  many  whole   things    in  20   tenths?     lu 
55  ?     In  60  ?     In  70  ?     In  80  ?     In  85  ?     In  95  ?     IP 
1001 


LESSON   VIII. 
Equal  parts  of  Numbers. 

1.  What  is  one  half  of  4? 

2.  What  part  of  1  is  one  third  of  2  ?* 

3.  What  part  of  1  is  one  fourth  of  3  ? 

4.  What  part  of  1  is  one  ninth  of  6? 

5.  What  part  of  1  is  one  fifth  of  4  1 

5.  How  many  times  1  is  one  sixth  of  8  ? 

7.  How  many  times  1  is  one  fifth  of  12? 

8.  Nine  are  how  many  times  7  ? 

9.  If  wheat    is   9    shillings    a   bushel,  how  ma\v 
bushels  can  you  buy  for  10  shillings?     Fur  15  shil 
lings  ?     For  20  shillings  ?     For  29  shillings  ? 

10.  Twelve  are  how  many  times  0  ? 

11.  What  part  of  1  is  one  twelfth. of  G  ? 

12.  What  part  of  1  is  one  tenth  of  4? 

13.  What  part  of  1  is  one  twelfth  of  9  ? 

4.  At  9  dimes  a  yard,  how  much  muslin 
can  you  buy  for  1  dime  ?  How  much  for  4 
dimes?  How  much  for  7?  How  much  for  9? 

*  SUGGESTION. — One  third  of  2  rnu«fcbe  twice  as  great  as  one 
third  of  1 :  one  third  of  one  is  one  third;  hence,  one  third  of  2 
is  two  thirds  of  1.  Similarly  for  each  example. 

OBSERVE  that  all  numbers  are  finally  expressed  in  terms  of 
the  unit  1. 


LES.  VIII.]  INTELLECTUAL    ARITHMETIC. 


81 


How    many    for   12  ?     How    many    for   18  ?     How 
many  for  20  ?     How  many  for  27  ? 

15.  If  coal  is  7  dollars  a  ton,  how  much   will   onf 
seventh  of  a  ton  cost?     5  sevenths  ?     14  sevenths? 

16.  If  you  have  30  dollars,   how  many  barrels  of 
flour  can  you  buy  at  5  dollars  a  barrel  ?     At  6  dol 
lars  a  barrel?     At  7  dollars'?     At  8   dollars?     A I 
9  dollars?     At  10  dollars? 

17.  Twenty -four  are  how  many  times  4  ?      How 
many  times  5?    8?    9?. 10?     11?     12? 

18.  Sixty  are  how  many  times  5  ?     7?     8  ?     4  ? 
9?     10?    %12? 

19.  Seventy  are  how  many  times  10?     7?     8? 
9?     11?     3? 

20.  Forty-two  are  how  many  times  8  ?     9  ?     10  ? 
12?     0?     7? 

21.  Fifty-six  are  how  many  times   12?     10?     9? 
8?-  11  ?     5? 

22.  Twenty-two*  are  how  many  times  5  ?     6  ?     7  ? 
8?     9?     10? 

23.  Forty-eight  are  how  many  times  7?     6?     5  ? 
4?     12?     11?     9? 

24.  Fifty-five    are  how  many   times  8  ?     9  ?     7  ? 
5?     3?     10?     11? 

25.  Forty-seven  are  how  many  times  5  ?     G  ?     9  ? 
10?     12?  '8?     7? 

26.  Forty-one  are  how  many  times  6  ?     7  ?     9  ? 
8?     5?     4? 

27.  If  cloth  is  7  dollars  a  yard,  how  many  yards 
can  you  buy  for  60  dollars  ?     How  much  could  you 
buy  at  8  dollars  a  yard  ?    At  10  dollars  ?   5  dollars  1 
11  dollars? 

28.  Eighty -two  are  how  many  times  4?     8?     9? 
12?     11? 

29.  Eighty-eight  are  how  many  times  7  ?     8  ?     0  (l 

10?    in'    12? 


82  INTELLECTUAL    ARITHMETIC.  [SEC.    VI. 

30.  Eighty-five  are  how  many  times  9  ?     8  ?     7  ? 
10?     11?     12? 

3 1 .  Eighty-seven  are  how  many  times  9  ?     8  ?     7  ? 
10?     11?     12? 

32.  Ninety  are  how  many  times  9  ?     8  ?   7  ?    10? 
11?     12? 

33.  Ninety-nine   are   how   many    times  9  \     11? 
12?     8?     10? 

34.  Seventy -two  are  how  many  times  6  ?     9  ?     7  ? 
8?     10?     if?     12? 

35.  Seventy-six  are  how  many  times  9?     8?     7! 
0?     10?     11?     12? 

36.  Seventy-eight  are  how   many   times   9?     8? 
7?     10?     12?     11? 

37.  At  7  dollars  a  barrel,  how  much  flour  can  you 
buy  for  2  dollars?     How  much  for  4  dollars?     How 
much  for  6  dollars?     For  8 dollars  ?     For  12  dollars? 
For  15  dollars  ?     For  20  dollars  ?     For  25  dollars? 
For  27  dollars  ?     For  32  dollars  »     For  35  dollars  ? 

38.  Fiftv-five  are    how   many  times  5?     6?     8? 
9?     10?    '12?     11  ? 

39.  Fifty-nine  are  how  many  times  8?     7?     6? 
9?     10?    *11?     12? 

40.  If  sugar  is  9  dollars  a  hundred  weight,  how 
many  hundreds  can  you  buy  for  3  dollars  ?     How 
much  for  6?     For  12?     For  18  dollars?     For  26? 
For  39  ?     For  65  ?     For  72  ?     For  92  dollars  ? 


LESSON    IX. 

Expressing  Fractions  by  Figures  and  Reading  them. 
1.  The  following  is  the  manner  of  expressing  frac 
tions  by  figures  : 

|     one  half.  \      one  seventh. 

£     one  third.  -J-      one  eighth. 

£     one  fourth.  £      one  ninth. 

£     one  fifth.  ^     one  tenth. 

J-     one  sixth.  ^    one  twelfth. 


LKS.  IX.]  INTELLECTUAL    ARITHMETIC. 

2.  What  is  the  figure  above  the  line  called  *     The 
numerator. 

3.  What  is  the  figure  below  the  line  called  ?     The 
denominator. 

4.  What  does  the  denominator  show  ? 

It  shows  into  how  many  equal  parts  the  whole 
thing  is  divided. 

5.  What  does  the  numerator  show? 
How  many  of  those  equal  parts  are  taken. 

6.  What  may  the  whole  thing  which  is  divided  be 
called  ?      Unity   or  one. 

7.  Is  one  apple  a  unit  ?     Is  one  peach  a  unit  1     Is 
one  dollar  a  unit  ?     Is  one  book  a  unit  1 

8.  In  the  fraction  -J.  into  how  many  equal  parts  is 
the  unit  divided  ]     Which  figure  is  the  numerator  1 
Which  the  denominator? 

9.  In  the  fraction  y1^,  into  how  many  equal  parts  is 
the  unit  divided  ?     Which  is  the  numerator  ?     Which 
the  denominator  ? 

10.  Read  the  following  fractions  : — 

-|      three  eighths.  1%     eight  thirteenths. 

three  sevenths.  yi     forty -one  sixteenths. 

five  ninths.  2f     two  and  3  eighths. 

six  twelfths.  5f     five  and  9  sevenths. 

T3-    seven  fifteenths.  6^  six  and  5  twelfths.* 

11.  In  the  fraction  -|,  into  how  many  equal  parts  is 
the  unit  divided  ?     How  many  of  these  parts  are 
taken  ? 

12.  In  the  fraction  £,  into  how  many  equal  parts  is 
the  unit  divided  1     How  many  parts  are  taken  ? 

*  SUGGESTIONS. — "When  the  numerator  is  less  than  the  deno 
minator,  the  fraction  is  called  &  proper  fraction. 

When  the  numerator  is  greater  than  the  denominator,  the 
fraction  is  called  an  improper  fraction. 

A  whole  number1  united  with  a  fraction,  is  called  a  mixfd 
number. 


V  V-- 

84  INTELLECTUAL    ARITHMETIC.  [<.jLC.  W 

13.  In  the  fraction  |,  into  how  many  equ^l  parts  is 
the  unit  divided  ?     How  many  parts  are  taken  ? 

14.  In  the  fraction  |,  into  how  many  equal  parts  is 
the  unit  divided  ?     How  many  parts  are  taken  ? 

15.  In  the  fraction  T6¥,  into  how  many  equal   parts 
is  the  unit  divided  1     How  many  parts  are  taken  ? 

16.  In  the  fraction  -j7^-,  into  how  many  'equal  parts 
is  the  unit  divided  ?     How  many  parts  are  take  n  1 

17.  In  the  fraction  y^-,  into  how  many  equal  parts 
is  the  unit  divided  1     How  many  parts  are  taken  ? 

18.  In  the  fraction  ||-,  into  how  many  equal  parts 
is  the  unit  divided  1     How  many  parts  are  taken  if 

19.  What  figures  express  one  third  of  one? 

20.  What  figures  express  five  sevenths  ? 

21.  What  figures  express  nine  twelfths? 

22.  What  figures  express  twelve  seventeenths? 

23.  What  figures  express  ten  thirty-sevenths  ? 

24.  What  figures  express  fifteen  sevenths  ? 

25.  What  figures  express  fourteen  twenty-ninths  ? 


LESSON   X. 

Fractional  Units. 

1.  What  is  the  unit  of  a  fraction  ? 

It  is  the  whole  thing  which  is   divided  into   equal 
parts. 

2.  What  is  each  equal  part  called  ? 
A  fractional  unit. 

3.  In  the  fraction  |  of  a  pound,  what  is  the  unit  of 
the  fraction  ? 

One  pound. 

W7hat  is  the  fractional  unit? 

One-eighth  of  a  pound. 

How  many  fractional  units  are  taken  ? 

Five. 

4.  In  the  fraction  f ,  what  is  the  unit  of  the  frac 
tion? 


LES.  X.]  INTELLECTUAL    ARITHMETIC. 


85 


The  abstract  or  simple  unit,  one. 

What  is  the  fractional  unit? 

One  fourth. 

How  many  fractional  units  are  taken  1 

Three. 

5.  What  is  the  fractional  unit  in  three  fourths? 
How  many  fractional  units  in  three  fourths'? 

6.  What  is  the  unit  of  the  fraction  f  of  a  dollar? 
What  is  the  fractional  unit  ? 

How  many  fractional  units  are  taken  ? 

7.  What  is  the  unit  of  the  fraction  |  ? 
What  is  the  fractional  unit? 

How  many  are  taken? 

8.  How  many  fractional  units  in  -}-J? 
What  is  the  unit  of  the  fraction  ? 
What  the  fractional  unit? 

9.  James  and  John  have  each  an  apple  of  the  same 
size.     James  cuts  his  into  two  equal  parts,  and  gives 
away  one  half.     John  cuts  his  into  three  equal  parts 
and  gives  away  two  of  the  pieces.     They  seek  to  find 
what  part  of  a  whole  apple  they  have  left.* 

*  SUGGESTION. — This  example  suggests  all  the  principles  em 
ployed  iii  fractions.  They  may  be  thus  stated  : 

1st.  That  something  regarded  as  a  whole,  called  unitv,  is  thi* 
primary  base  of  every  fraction  :  and 

2d.  That  one  of  the  equal  parts  of  unity,  called  the  fractional 
unit,  is  the  second  base  of  any  fractional  number. 

From  the  nature  of  Division  and  Multiplication,  we  see  : 

1st.  That  the  fractional  unit  is  as  many  times  less  than  unity 
as  there  are  units  in  the  denominator : 

2d.  That  the  numerator  shows  how  many  fractional  units  ai<* 
taken. 

8d.  If  the  numerator  be  multiplied  by  any  number,  the  num 
ber  of  fractional  units  will  be  increased  as  many  times  as  there 
are  units  in  the  multiplier. 

4th.  If  the  numerator  be  divided  by  any  number,  the  number 
of  fractional  units  will  be  diminished  as  many  times  as  there 
are  units  in  the  divisor. 

8 


INTELLECTUAL    ARITHMETIC.  [SEC.  VL 

Now,  says  James,  if  I  cut  my  half  into  3  equal 
parts,  each  part  will  be  three  times  less  than  before; 
that  is,  each  part  will  be  one  sixth  of  the  entire  apple, 
but  I  shall  have  three  times. as  many  parts,  ^o  that  I 
shall  still  have  half  my  apple. 

Now,  says  John,  if  I  cut   my  third   into  2  equal 

arts,  each  part  will  be  two  times  less  than  before; 

hat  is,  it  will  be  one-sixth  of  the  entire  apple  ;   but  1 

shall  have  twice  as  many  parts,  so  that  I  still  have 

cne  third  of  the  apple. 

Now,  says  John  to  James,  if  you  have  3  sixths  and 
I  have  2  sixths,  together  we  must  have  5  sixths  of  a 
whole  apple. 

10.  In  the  fractions  •£  and  ^-,  what  is  the  fractional 
unit  of  the  first  1     What  of  the  second  ? 

How  can  you  reduce  them  to  the  same  fractional 
unit? 

By  multiplying  the  numerator  and  denominator  of 
the  first  by  3,  and  of  the  second  by  2. 

What  is  the  fractional  unit  after  the  multiplication  ? 

How  many  fractional  units  are  taken  in  each  frac 
tion  ?  How  many  in  both  ? 

11.  How  can  you   reduce   fractions  to  the  same 
fractional  unit? 

By  multiplying  the  numerator  and  denominator  of 

5th.  If  the  denominator  be  multiplied  by  any  number,  the 
fractional  unit  will  be  diminished  as  many  times  as  there  are 
units  in  the  multiplier. 

Cth.  If  the  denominator  be  divided  by  any  number,  the  value 
of  the  fractional  unit  will  be  increased  as  many  times  as  there 
are  units  in  the  divisor. 

By  combining  3  and  5,  we  see  : 

7 th.  That  if  the  numerator  and  denominator  be  multiplied  by 
the  same  number,  the  value  of  the  fraction  will  not  be  changed 

And  by  combining  4  and  6,  we  see : 

8th.  That  if  the  numerator  and  denominator  be  both  divided  "by 
the  tame  number,  the  value  of  the  fraction  will  not  be  changed. 


LES.   X.]  INTELLECTUAL    ARITHMETIC.  87 

each  fraction  by  such  a  number  as  shall  make  the 
•denominators  the  same  in  all. 

12.  Reduce  4-  and  £,  to  the  same  fractional  unit?* 
If  we  multiply  the  numerator  and  denominator  of 

th(j  first  fraction  by  4,  we  have  T\ ;  and  if  we  multi 
ply  the  second  by  3,  we  have  r%,  in  which  the  frac 
tional  unit  is  ^V  m  both. 

13.  Reduce  J-  and  |  to  the  same  fractional  unit. 
After  reduction,  what  is   the   fractional  unit,  and 

how  many  fractional  units  in  both? 

14.  Reduce  £  and  ^  to  the  same  fractional  unit. 
After  reduction,   what  is   the  fractional  unit,  and 

how  many  fractional  units  in  both? 

15.  Reduce  1,  f,  and  -|,  to  the  same  fractional  unit. 
After  reduction,  how  many  fractional  units  in  each  1 

16.  Reduce  |  and  f  to  the  same  fractional  unit. 
After  reduction,  what  is  the  fractional  unit? 
How  many  in  each  ? 

17.  Reduce  -|,  J,  and  |,  to  the  same  fractional  unit. 

18.  Reduce  |,  f,  and  f ,  to  the  same  fractional  unit. 

19.  Reduce  ±,  |,  and  |,  to  the  same  fractional  unit. 

20.  Reduce  |  and  £  to  the  same  fractional  unit. 

21.  Reduce  ±  anr}  |.  to  the  same  fractional  unit. 

22.  Reduce  -J  and  -|  to  the  same  fractional  unit, 

23.  Reduce  J ,  f,  and  f ,  to  the  same  fractional  unit. 

24.  In  V5  how  many  units  1  ? 

25.  In  6g7  how  many  units  1  ? 

26.  In  4g5  how  many  units  1  ? 

27.  In  3^6  how  many  units  1  ? 

28.  In  Y  how  many  units  1  ? 

29.  In  2~*  how  many  units  1  ? 

*  SUGGESTIONS. — There  are  but  three  operations  which  cLang 

the  units  of  numbers :     They  are, 

1st.  To  change  integral  to   fractional  units: 

2d.  To  change  fractional  to  integral  units :  and 

3d.  To  change  from  one  fractional  unit  to  another. 

The  first  two  were  fully  explained  in  the  first  seven  lessons 

of  thie  section :  and  the  third  is  treated  of  in  this. 


INTELLECTUAL    ARITHMETIC.  [SEC.  VL 

SO.  In  2ij7  how  many  units  1  ? 

31.  In  3¥9  how  many  units  1  ? 

32.  In  5g4  how  many  units  1  1 

33.  In  7g5  how  many  units  1  1 

34.  How  many  units  1  in  357  1 

35.  How  many  units  1  in  4y9  ? 

36.  How  many  nnits  1  in  6T3  ? 

37.  How  many  units  1  in  4y9  ? 

38.  How  many  fourths  in  1  half? 

39.  How  many  fourths  in  2  halves] 

40.  How  many  fourths  in  6  halves? 

41.  How  many  thirds  in  12  sixths? 

42.  How  many  sixths  in  1  third  1 

43.  How  many  sixths  in  4  thirds  ? 

44.  How  many  sixths  in  1  half? 

45.  How  many  sixths  in  2  halves? 
4G.   How  many  eighths  in  1  fourth? 

47.  How  many  eighths  in  1  half? 

48.  How  many  eighths  in  2  fourths  ? 

49.  How  many  eighths  in  3  fourths? 

50.  How  many  ninths  in  1  third  ? 

51.  How  many  ninths  in  2  thirds? 

52.  How  many  ninths  in  6  thirds? 

53.  How  many  tenths  in  1  fifth  ? 

54.  How  many  tenths  in  1  half? 

55.  How  many  twelfths  in  1  third? 

56.  How  many  twelfths  in  1  half? 

57.  How  many  twelfths  in  1  quarter  ? 

58.  How  many  sixteenths  in  1  eighth  ? 

59.  How  many  sixteenths  in  1  quarter  ? 
00.  How  many  eighteenths  in  1  ninth  ? 
OL  How  many  eighteenths  in  1  half? 

62.  How  many  eighteenths  in  2  thirds?  . 

63.  How  many  twentieths  in  3  halves? 

64.  How  many  twentieths  in  3  quarters? 

65.  How  many  twentieths  in  5  fourths? 

66.  How  many  twentieths  in  3  fifths  ? 


LE8.   XI.]  INTELLECTUAL    ARITHMETIC. 

67.  How  many  halves  in  T8g  ?     In  -^  *?* 

68.  How  many  halves  in  f|  1     In  £  ? 

69.  How  many  thirds  in  |  'I     In  ^1 

70.  How  many  thirds  in  T8T?     In  r%? 

71.  How  many  thirds  in  ¥9T?     In  f-f  ? 

72.  How  many  fourths  in  f f  1     In  £f  ? 

73.  How  many  fourths  in  f £  ?     In  r4^- 1 

74.  How  many  fourths  in  ||?     In  ^|.? 

75.  How  many  fourths  in  \^.     In  ^-|] 

76.  How  many  fifths  in  T%  •?     In  ^  ] 

77.  How  many  fifths  in  f §  ?     In  ^- 1 

78.  How  many  sixths  in  |£  1     In  T8-g-  ] 

79.  How  many  sixths  in  ^-fl     In  ||  ? 
an  How  many  sevenths  in  -J-J  ?     In  -|-| 

How  many  sevenths  in  |^- 1     In  T^- 


80. 
81. 


LESSON  XI. 
Value  of  Numbers. 

1.  What  is  the  base  of  a  number  ? 

The  primary  base  of  every  number  is  the  UNIT  ONE. 

2.  What  is  an  integer,  or  whole  number? 

It  is  a  number  which  contains  the  unit  one  an  exact 
number  of  times.  Thus,  three,  four,  five,  &c.,  are 
integers,  or  whole  numbers. 

3.  What  is  a  fractional  number  ? 

It  is  a  number  which  expresses  one  or  more  of  the 
equal  parts  of  unity.  Thus,  -J,  f,  f,  &c.,  are  fractions. 

4.  What  expresses  the  value  of  a  number  either 
integral  or  fractional  1 

The  number  of  times  which  it  contains  the  unit  one. 

5.  How  many  times  does  five  contain  one  1 

6.  How   many    times   is  five  greater   than  one  ? 
Why  ] 

*  SUGGESTIONS. — Divide  the  numerator  and  denominator  by 
such  a  number  as  will  give  the  required  fractional  unit 
8* 


90  INTELLECTUAL    ARITHMETIC.  [SEC.  VI. 

7.  How  many  times  is  six  greater  than  one  ?   Why  'I 

8.  How    many  times  is   eight  greater  than  one? 
Why? 

9.  How  many  times  does  one  half  contain  the  unit 
one? 

One-half  time. 

10.  How  many  times  is  one  half  less  than  one  ? 
Two  times. 

11.  What  expresses  the  value  of  a  fraction  ? 
The  number  of  times  which  a  fraction  contains  1, 

12.  What  is  the  value  of  one  half?     Why  1 

13.  What  is  the  value  of  two  halves  ?     Why  ? 

14.  What  is  the  value  of  three  halves]     Why  ? 

15.  What  is  the  value  of  four  halves  ?     Why  ? 

16.  What  is  the  value  of  one  third  1     Why  ? 

17.  What  is  the  value  of  three  thirds  ?     Why  ? 

18.  What  is  the  value  of  six  thirds  ?     Why  ? 

19.  How  do  you  find  the  value  of  an  improper 
fraction  1* 

By  dividing  the  numerator  by  the  denominator. 

20.  If  there  is  a  remainder,  what  do  you  do  with  it? 
Write  the  denominator  under  it,  and  annex  the 

fraction  to  the  integral  number. 

21.  What  is  the  value  of  the  fraction  seven  thirds? 

22.  What  is  the  value  of  nine  fourths  ?     Of  seventy- 
two  twelfths  ?     Of  one  hundred  eighths  ?     Of  ninety- 
four  ninths  ? 

23.  What  is  the  value   of  seventy-five  twelfths  f 
Of  sixty-seven  eighths?     Of  eighty-nine   elevenths? 
Of  one  hundred  and  twenty  twelfths'? 

*  ANALYSTS. — Since  the  numerator  shows  how  many  fractional 
units  are  taken,  and  the  denominator  how  many  fractional  units 
make  1,  it  follows,  that  the  numerator  divided  by  the  denominator 
will  show  how  many  units  1,  there  are  in  the  fraction. 

Impress  the  pupil,  constantly,  that  every  number,  whether 
integral  or  fractional,  must  be  compared  with  the  unit  one. 
Also,  that  the  value  of  any  number  is  expressed  by  the  number 
of  times  which  it  contains  the  unit  one. 


LKS.    Xl.J  INTELLECTUAL    ARITHMETIC.  91 

24.  When  is  a  fraction  said  to  be  in  its  lowest 
terms  ? 

A  fraction  is  said  to  be  in  its  lowest  terms  when 
there  is  no  number  except  1  that  will  divide  both 
the  numerator  and  denominator  without  a  remainder. 

25.  How  may  a  fraction  be  reduced  to  its  lowest 
*erms? 

By  dividing  both  the  numerator  and  denominator 
by  the  same  number.* 

20.  Is  the  fraction  f  in  its  lowest  terms  1     Why  ? 

27.  What  fraction  expresses  the  lo  west  terms  of  |  7 

28.  What    fraction    expresses    the  lowest  terms 

GfTV? 

29.  What    fraction    expresses    the    lowest    terms 

of  TV? 

30.  What    fraction    expresses    the    lowest   terms 
of  T\  ? 

31.  What  are  the  lowest  terms  of  the  fraction  ^  ? 

32.  What  are  the  lowest  terms  of  the  fraction  -|  ? 

33.  What  are  the  lowest  terms  of  the  fraction  f%  ? 

34.  What  are  the  lowest  terms  of  the  fraction  ¥8T? 

35.  What  are  the  lowest  terms  of  the  fraction  -/^l 
30.   What  are  the  lowest  terms  of  the  fraction  ^f  ? 

37.  Reduce  the  fraction  ^-|  to  its  lowest  terms. 

38.  What  fraction  will  express  the  lowest  terms 
of  f|? 

39.  What  fraction  will  express  the  lowest  terms 

rfH* 

40.  What  fraction  will  express  the  lowest  terms 


SUGGESTION.  —  Reduce  every  fraction  to  its  lowest  terms  before 
performing  any  other  operation.  Reduce  also  every  anneer  it 
lis  lowest  terms. 


02  INTELLECTUAL    ARITHMETIC.  [SEC.  VI* 

LESSON    XII. 
Adding  Fractional  Units. 

James  and  John  have  each  an  apple  of  the  same 
size.  James  cuts  his  into  4  equal  parts  and  gives 
away  2  parts.  John  cuts  his  into  5  equal  parts  and 
gives  away  3  parts.  They  seek  to  find  what  part  of 
an  apple  each  has  left,  and  what  part  both  together 
have  ? 

Now,  says  James,  if -I  cut  each  of  my  fourths 
into  5  equal  parts,  I  shall  have  10  parts  ;  that  is,  10 
twentieths  of  an  apple:  and  says  John,  if  I  cut  each 
of  my  fifths  into  4  equal  parts,  I  shall  have  8  parts; 
that  is,  8  twentieths  of  an  apple ;  hence,  James  has 
10  twentieths,  and  John  8  twentieths,  and  together 
they  have  18  twentieths. 

2.  What  is   the  unit  of  the  fraction  %$  1     What 
is  the  fractional  unit  in  1$  twentieths  of  an  apple  1 
How    many    fractional    units    are    taken  ?      If  the 
numerator  and  denominator  be   each   divided  by  2, 
what  does   the  fraction    become  1      What,    then,    is 
each  fractional  unit  ?     How  many  are  taken  ? 

3.  What  is  necessary   in  order  that  two  or  more 
fractions  may  be  added  together. 

That  they  have  the  same  integral  unit,  and  the 
same  fractional  unit. 

4.  When  no  integral  unit  is  named,  what  unit  is 
understood? 

The  abstract  unit  one. 

5.  Have  the  fractions  \  and  f  the  same  integral 
unit?     What  is  it?     Have  they  the  same  fractional 
unit?     What  is  it? 

6.  How  many  fractional  units  in  the  first?     How 
many  in  the  second  ?     How  many  in  both  ? 

7.  What  is  the   sum  of  the  fractional  units  in  \ 
and  1? 

8.  What  is  the  sum  of  f  and  f  ? 


LES.   XII.  J  INTELLECTUAL    ARITHMETIC.  93 

9.  What  is  the.  sum  of  £,  f  ,  and  f  ? 

10.  What  is  the  sum  of-£,  f,  and  f  ? 

11.  What  is  the  sum  of  |-,  f,  and  f  ? 

12.  What  is  the  sum  of  f  ,  f  ,  •£  ,  and  f  ? 

13.  What  is  the  sum  of  |,  f,  and  |? 

14.  What  is  the  sum  of  f,  y,  and  |?  . 

15.  What  is  the  sum  of  J,  f,  f,  and  }? 

16.  What  is  the  sum  of  |,  f,  f,  and  f  ? 

17.  What  is  the  sum  of  f,  f,  f,  and  f  ? 

18.  What  is  the  sum  of  f  ,  |,  f  ,  and  f  ? 

19.  What  is  the  sum  of  f,  |,  f,  and  |  ? 

20.  What  is  the  sum  of  f,  -|,  -f  ,  and  1  | 

21.  What  is  the  sum  of  f  .  f  ,  f  ,  and  |  ? 

22.  What  is  the  sum  of  y3,  -f-,  f,  and  |  ? 

23.  What  is  the  sum  of    , 

24.  What  is  the  sum  of 


D, 

25.  What  is  the  sum  of  i,  |,  |,  f 

26.  What  is  the  sum  of  f,  f,  |,  and  f  f 

27.  What  is  the  sum  of  f,  f  ,  f,  and  \  ]    , 

28.  What  is  the  sum  of  f,  -f-,  J,  and  f  ? 

29.  What  is  the  sum  of  f  ,  |,  f  ,  and  LO  ? 
80.  Add  together  £  and  i. 

First,  I  is  equal  to  f.  and  -J  is  equal  to  |  ;  then  | 
plus  f  are  equal  to  -|. 

31.  What  is  the  sum  off  and  |  ? 

32.  What  is  the  sum  of  f  and  1  1     (Reduce  to 
thirtieths.) 

33.  What  is  the  sum  of 

34.  What  is  the  sum  of 

35.  What  is  the  sum  of 

36.  What  is  the  sum  of 

37.  What  is  the  sum  of 

38.  What  is  the  sum  of  f  and 
89.    What  is  the  sum  of  ^  and  |  ? 

40.  What  is  the  sum  of  |  and  f? 

41.  What  is  the  surn  of  |  and  |] 


94  INTELLECTUAL    ARITHMETIC.  [SEC.  Vi 

42.  What  is  the  sum  of '2|  and  3-}  and  5-^  ?* 
13.   What  Is  the  sum  of  2'  and  4f  and  6£? 

44.  What  is  the  sum  of  if  and  0|-  and  72%? 

45.  What  is  the  sum  of  4|  and     f  and  2^>3y  ? 

46.  What  is  the  sum  of  (,)|  and    f  and  7/g-  ? 
47.'  What  is  the  sum  of  5f  and  4f  and  ^  ? 

48.  What  is  the  sum  of  9£  and  7f  and  ^  ? 

49.  What  is  the  sum  of  5£  and  4^  and  8-3%  1 

50.  What  is  the  sum  of  6|  and  8r%  and  fO/y? 

51.  What  is  the  sum  of  5f  and  8-^  and  2^-1 

52.  What  is  the  sum  of  5  and  2J  and  3^  ? 

QUESTIONS. 

1  John  buys  a  top  for  one  sixth  of  a  shilling,  a 
stick,  of  candy  for  one  twelfth  of  a  shilling,  and  a 
piece  of  india-rubber  for  one  third  of  a  shilling  :  what 
does  the  whole  cost  him?f 

2.  James  pays  ^  of  a  dollar  for  a  pair   of  gloves, 
and  |  of  .a  dollar  for  a  handkerchief:  how   much  do 
they  cost  him  ? 

3.  Nancy  buys  a  work-box   for  |-  of  a  dollar,  a 
pair  of  gloves  for  f  of  a  dollar,  and  a  comb  for  T2^  of 
a  dollar:  how  much  do  they  all  cost? 

4.  Jane  buys  a  yard  of  ribbon  for  |-  of  a  dollar,  a 
gold  pin  for  f  of  a  dollar,  and  an  inkstand  for  T3^  of 
a  dollar  :  how  much  did  she  pay  in  all? 

5.  William  buys  a  kite   for  |  of  a  dollar,  and  a 
string  for  T^'of  a  dollar:  how  much  did  he  pay  ? 

6.  Three  ducks  cost  f  of  a  dollar,  two  fowls  J  of  a 
dollar,  and  two  geese  y9^  of  a   dollar  :    what   is  the 
entire  cost? 

*  SUGGESTIONS. — Add  the  integer  numbers    separately,  and 
unite  the  sum  to  the  sum  of  the  fractional  parts. 
f  ANALYSIS. —  -g-  of  a  shilling  is  -fj  of  a  shilling. 
y1^  of  a  shilling  is  y  ^ 
•5-    of  a  shilling  is  =p  2 
nence  the  whole  costs  is    -     -     -     ^  °f  a  shilling. 


LES.  XII. J  INTELLECTUAL    ARITHMETIC.  95 

7.  Three  sheep  cost  y   of  a  dollar,  a  calf  f  of  a 
dollar,  and  a  lamb  J  of  a  dollar :  what  is  the  entire 
cost? 

8.  Three  yards  of  cloth  cost  |  of  a  dollar,  a  hand 
kerchief  \   of  a  dollar,  and  a  pair  of  gloves  j  of  a 
dollar:  what  is  the  entire  cost? 

9.  A  father  paid  3  quarters  of  a  dollar  for  his  own 
breakfast,  one  third  of  a  dollar  for  his  son's,  and  a 
quarter  of  a  dollar  for  his  daughter's:    how  much 
did  he  pay  in  all  ? 

10.  A  merchant  sold  3-|  yards  of  cloth  from  one 
piece,  2J  yards  from  another,  and  5|-  yards  from  an 
other  :  how  much  did  he  sell  in  all  ?* 

11.  If  a  turkey  costs  |  of  a  dollar,  a  goose  -J-  of  a 
dollar,  and  2  chickens  \  of  a  dollar :  how  much  will 
the  whole  cost? 

12.  James  spends  6-J-  cents  for  candy,  12-*-  cents 
for  a  top,  and  5^  cents  for  a  slate  :  how  much  does 
he  spend  in  all  ? 

13.  A  man  travelled  2J  miles  the   first  hour,  3|- 
miles  the  second,  arid  4|  the  third  :  how  for  did  he 
travel  in  the  three  hours  ? 

14.  What  amount  was  paid  for  4  weeks  board,  the 
board  for  the  first  week  being  51  dollars,  for   the 
second  3-g-  dollars,  for  the  third  4^  dollars,  and  for  the 
fourth  4f  dollars  ? 

15.  A  man  paid  |  of  a  dollar  for  a  breakfast,  1|  of 
-  a  dollar  for  dinner,  -|  of  a  dollar  for  supper,  and  |  of  a 

dollar  for  lodging:  what  did  he  pay   for  the  days' 
entertainment? 

1 6.  J  ane  paid  3|  cents  for  tape.  6|  cents  for  needles, 
«nd  4-j-J  cents  for  ribbon  :  how  much  did  she  pay  in 
all? 

17.  If  5  yards  of  muslin  cost  J  of  .a  dollar,  9  pairs 
of  stockings  2|  dollars,  and  2  pairs  of  boots  9|-  dol 
lars,  what  will  be.  the  whole  cost  ? 


06  INTELLECTUAL    ARITHMETIC.  [SEC.    VI. 

18.  A  laborer  sawed  weed  for  4  hours,  and  was  to 
have  86  cents  a  cord.  The  first  hour  he  sawed  |  of  a 
cord,  the  second  f  of  a  cord,  the  third  £  of  a  cord, 
and  the  4th  hour  ^  of  a  cord:  how  much  ought  he 
to  receive  ? 


LESSON    XIII. 

Subtracting  Fractions. 

1.  What  is  necessary  in  order  that  one  fraction 
may  be  subtracted  from  another  ? 

That  both  fractions  have  the  same  integral  and  the 
same  fractional  unit. 

2.  From  -£  subtract  y. 

Since  the  integer  unit  is  the  abstract  unit  one,  and 
the  fractional  unit  in  both,  is  1,  the  difference  is  found 
by  subtracting  2  sevenths  from  4  sevenths,  which 
leaves  2  sevenths. 

3.  What  is  the  difference  between  -f-%  and  T5^  ? 

4.  What  is  the  difference  between  T8^  and  T\? 

5.  What  is  the  difference  between  14  an(j  j^  ? 

6.  What  is  the  difference  between  -J-J  and  j~  1 

7.  What  is  the  difference  between  ^f-  and  if  1 

8.  What  is  the  difference  between  y|  and  TJ  ? 

9.  What  is  the  difference  between  T|-  and  T|  ? 

10.  What  is  the  difference  between  ffi  and  i|-? 

11.  What  is  the  difference  between  |-f-  and  |^  ? 

12.  What  is  the  difference  between  i|-  and  -fj  ? 

13.  What  is  the  difference  between  J-^-  and  -^y  1 

14.  What  is  the  difference  between  ^  and  -J-?     Re 
duce  both  to  the  fractional  unit  one  sixth. 

One  half  is  equal  to  -|,  and  one  third  to  |  ;  hence, 
their  difference  is  equal  to  -J-. 

15.  What  is  the  difference  between  -J  and  -§-  ? 

16.  What  is  the  difference  between  -|  and  y2^  ? 

17.  What  is  the  difference  between  T5T  and  1  ? 


LES.  XIII.]  INTELLECTUAL    ARITHMETIC.  97 

18.  What  is  the  difference  between  }-|  and  \  ? 

19.  What  is  the  difference  between  |-  and 

20.  What  is  the  difference  between  -^  and 

21.  What  is  the  difference  between  -f-±  and  - 

22.  What  is  the  difference  between  -fj-  and  ||  ? 

23.  From  2|  take  1 J* 

24.  What  is  the  difference  between  5^  and  2-|? 

25.  What  is  the  difference  between  31  and  2£? 

26.  What  is  the  difference  between  1|  and  -| '? 

27.  What  is  the  difference  between  2^  and  1|  ? 

28.  What  is  the  difference  between  l|-  and  2^1 

QUESTIONS. 

1.  If  you  give  $  of  an  orange  to  one  boy  and  J  to 
another,  how  much  more  do  you  give  to  one  than  to 
the  other  ? 

2.  If  I  have  f  of  a  dollar  and  give  $  dollar  for  a 
knife,  how  much  would  I  have  left? 

3.  William  had  f  of  a  dollar  and  gave  £  of  a  dollar 
to  a  beggar,  how  much  had  he  left  ? 

*  ANALYSIS. — First:  2j  are  equal  to  y,  equal  to  f  £ ;  and 
Ij;  are  equal  to  f-  equal  f  f  ;  hence,  the  difference  is  J|. 

It  is  generally  best  to  subtract  the  integral  and  fractional  iiuin 
bers  separately :  thus,  in  example  25,  3  j,  less  2^-,  we  may  say 
8  less  2  equals  1 ;  -|  less  -g-  equals  f,  equals  ^  :  hence,  the  tru* 
difference  is  1^-. 

If  the  fractional  part  of  the  subtrahend  is  of  greater  value 
than  the  fractional  part  of  the  minuend,  take  one  of  the  Integra; 
uuits  of  the  minuend  and  add  it  to  th«  fractional  part,  and  thei- 
subtract. 

Thus,  m  example  24,  f  is  greater  than  i  :  hence,  we  take  1 
unit  from  5  which  added  to  ^  makes  f ;  and  ^  less  |  leaves  -f^ , 
then  adding  1  to  the  next  figure  in  the  subtrahend,  which  is  the 
same  A3  taking  1  from  the  minuend,  we  have  3  from  5  leaves  2; 
therefore,  the  difference  between  5j  and  2|  is  2T\. 


98  INTELLECTUAL    ARITHMETIC.  [SEC.  VI. 

4.  B  travels  f  of  a  mile  in  the  same  time  that  C 
travels  J:  which  travels  the  farthest  and  how  much1? 

5.  A  merchant  sells  -J-  of  a  barrel  of  sugar  from  a 
barrel  |-  full :   what  part  was  there  left  ? 

6.  A  tailor  cut  -J  of  a  yard  of  cloth  from  a  piece 
containing  If  yards:  how  much  was  there  left? 

7.  John  pays  |-  of  a  shilling  for  a  knife,  and  -J-  of  a 
shilling  for  a  top  :  for  which  does  he  pay  the  most  3 
How  much  ? 

8.  Four  pounds  of  tea  cost  ^  dollars,  and  twenty 
pounds  of  sugar  jf :   which   costs   the   most  ?     How 
much  ? 

9.  A   farmer  buys    a  calf,  for  which  he  pays  J 
dollars,  and  a  lamb,  for  which  he  pays  g   dollars: 
for  which  does  he  pay  the  most?     How  much  ? 

10.  James'  shoes  cost    l£    dollars,   and    his   vest 
J-J-  dollars  :  what  is  the  difference  of  their  cost  ? 

11.  A  man  earned  in  4  days  of  a  week,  5y  dollars, 
and   paid   1^   dollars  for   his  board,  the  three  other 
days:  how  much  should  he  receive? 

12.  From   a  piece  of  cloth  which  was  12|   yards 
long,  of  yards  is  cut :  how  many  yards  are  left? 

,13.  If  from  a  box  of  sugar  containing  18-|  pounds, 
Gf  pounds  are  taken,  how  much  will  be  left? 

14.  A  grocer  bought  16|  bushels  of  beans,  and  after 
selling  5f  bushels,  how  many  has  he  left  ? 

15.  Jane  is  15§  years  old,  and  Nancy 'is  9|-  years 
old  :  how  many  years  is  Jane  older  than  Nancy  ? 

16.  A  draper  cuts  5y  yards  of  cloth  from  a  piece 
21  {I  yards  long:  how  much  is  left? 

17.  A  grocer  purchases  a  box  of  eggs,  for  which  he 
paid  3J  dollars,  and  sold  them  for  5f  dollars :  how 
much  did  he  make? 

18.  A  grocer  bought  a  pair  of  chickens  for  f  of  a 
dollar,  and  sold  them  for  |-  dollars  :  did  he  make  or 
lose,  and  how  much  ? 


LBS.  XIV.]  INTELLECTUAL    ARITHMETIC.  99 

LESSON   XIV. 
Multiplication  of  fractions. 

I.  How  many  are  3  times  §  ] 

Here  the  fractional  unit  is  one  third,  and  there  are 
two  fractional  units  in  the  expression,  which  being 
taken  3  times,  gives  six  thirds,  or  2  for  the  product. 
,     2.  How  many  are  5  times  6%-1*     8  times  ~£ 

3.  How  many  are  2  times  1  ?     6  times 

4.  How  many  are  8  times  |  ?     7  times 

5.  How  many  are  4  times  f?     3  times 

6.  How  many  are  7  times  ¥4T  ?     6  times  2£ 

7.  How  many  are  3  times  2^-  ?     5  times  34- 

8.  How  many  are  8  times  9j  ?     4  times  6f  ] 

9.  How  many  are  4  times  6-j-  ?     3  times  9^-  ? 
10.  How  many  are  10  times  5|  ?     9  times  G T^  1 

II.  How  many  are  7  times  6y  ]     9  times  9  ^  '? 

12.  How  many  are  5  times  3|- 1     8  times  6-f-  ] 

13.  How  many  are  7  times  4|  ?     0  times  3f  ? 

14.  If  the  denominator  of  a  fraction  be  divided  by 
a  whole  number,  how  will  the  fraction  be  affected  1 

It  will  be  increased  as  many  times  as  there  are 
units  in  the  divisor.  (See  Analysis,  page  86.) 

How  then  may  a  fraction  be  multiplied  by  a  whole 
number. 

Either  by  multiplying  the  numerator,  or  dividing 
the  denominator  by  the  multiplier. 

15.  What  is  the  product  of  |  by  2?  both  ways. 
1C.  What  is  the  product  of  f  by  4?  both  ways. 

17.  What  is  the  product  of  |  by  7?  both  ways. 

18.  What  is  the  product  of  |  by  4?  both  ways. 

19.  What  is  the  product  of  |  by  9  ?  both  ways. 

*  SUGG  <;S'TION. — Let  the  fractional  and  integral  numbers  b'» 
multiplied  separately  :  thus,  5  times  3  sixths  are  15  sixths,  equal 
to  2  and  a  half;  and  5  times  6  are  30 ;  to  which  add  :;.£,  giving 
32 i  for  the  product. 


100  INTELLECTUAL    ARITHMETIC.  [SEO.    VI. 

20.  What  is  the  product  of  ^  by  8  1  both  ways. 

21.  What  is  the  product  of  T<L  by  3  ?  both  ways. 

22.  What  is  the  product  of  jfby  8?  both  ways. 

23.  What  is  the  product  of  \  by  4;  both  ways. 

24.  What  is  the  product  of  T4T  by  5  ?  both  ways. 

25.  If  |  be  multiplied  by  4,  what  is  the  product  ? 

26.  What  is  the  cost  of  $  of  a  yard  of  cloth  at  | 
dollars  a  yard?     What  is  £  of  |1*     What  part  is 
it  of  1  ?f  v 

27.  What  is  the  cost  of  f-  boxes  of  raisins,  at  | 
dollars  a  box  ? 

28.  What  part  of  1  is  ?  of  f  1     What  is  1  of  f  ? 

29.  What   is   the   value  of  f  of  -f  ?    What  par 

of  n 

30.  What  is  the  value  of  2^  of  1}  1 

31.  What  is  the  value  of  f  of  f  of  £1 

32.  What  is  the  value  of  |  of  2  ? 

33.  How  many  times  1  is  3£  of  f  ? 

34.  How  many  times  1  is  2f-  of  l£? 

35.  How  many  times  1  is  3£  of  2? 

36.  How  many  times  1  is  5 J  of  6? 

37.  How  many  times  1  is  |  of  £  of  4  ? 

38.  How  many  times  1  is  |-  of  -|  of  |? 

39.  How  many  times  1  is  |  of  2J  of  3  ? 

40.  How  many  times  1  is  -|  of  ^  of  7. 

41.  How  many  times  1  is  |-  of  1J? 

*  The  word  OF  signifies  multiplication. 

f  ANALYSIS. — If  the  price  per  yard,  |  of  a  dollar,  be  multiplied 
by  f  5  the  number  of  yards,  the  product  will  be  the  cost. 

Now,  if  1  yard  of  cloth  cost  f  of  a  dollar,  ^-  of  a  yard  will 
«ost  one  fourth  as  much,  that  is,  ^4  of  a  dollar,  and  J  will  cost 
8  times  as  much  as  ^ ,  that  is,  J| ,  or  f  of  a  dollar.  Hence : 

The  product  of  two  fractions  is  found  by  multiplying  the 
itumerators  together,  and  the  denominators  together. 


LES.  XIV.]  INTELLECTUAL    ARITHMETIC.  101 

QUESTIONS. 

1.  What  will  5  yards  of  cloth 'ct>st,  at  $lf  a  /acci? 

2.  What  will  4  pounds  of- tea  cost,  at  5T4^  shillings 
a  pound  ?  !  :      •  •  .    '         '  S  f  * 

3.  What  will  7  dozen  of  apple's  cost,  at  li~-  cents 
a  dozen  ? 

4.  What  will  9  dozen  of  oranges  cost,  at  3J  shil 
Jings  a  dozen  ? 

5.  What  will  5  pairs  of  shoes  cost,  at  $2|-  a  pair? 

6.  What  will  10  hats  cost,  at  3T7^  dollars  apiece] 

7.  What  will  4f  yards  of  cloth  cost,  at  $7  a  yard? 

8.  What  will  be  the  cost  of  9f  yards,  at  $5  a 
yard  ? 

9.  What  will  12  pounds  of  coffee  cost,  atllf  cents 
a  pound  ? 

10.  What  will  9  sheep  cost,  at  $2f  a  head  ? 

11.  What  will  8  calves  cost,  at  $4J  a  head  1 

12.  What  will  11  quills  cost,  at  1^-  cents  apiece  ? 

13.  What  will  10  yards  of  carpet  cost,  at  $2f  ? 
yard  ? 

14.  What  will  9  pairs  of  gloves  cost,  at  $f  a  pair  ? 

15.  What  will  G  pairs  of  fowls  cost,  at -f-  of  a  dolla? 
a  pair  1 

16.  What  will  9  pairs  of  boots  cost,  at  5£  dollars 
a  pair  1 

17.  What  Will  be  the  cost  of  6  yards  of  cloth,  at 
4|-  dollars  a  yard  ? 

18.  What  is  the  product  of  2£  off  by  £? 

19.  Bought  f  of  If  yards  of  cloth  at  f  of  £  of  4 
dollars  a  yard:  what  did  it  come  to  ] 

20.  If  raisins  are  2J  dollars  a  box,  what  will  be  the 
Dst  of  16  boxes'? 

21.  What  will  be  the  cost  of  11  hats,  if  1  hat  cost 
3|  dollars  ? 

22.  If  1  pair  of  shoes  cost  $2|-  dollars,  what  will 
be  the  cost  of  7  pairs  ] 

9* 


INTELLECTUAL    ARITHMEIC.  [SEC.  IV. 

23.  What  will  12  -penknives  cost,  at  $1-J  apiece  ? 
.24.  What  .will.  3,  To.  of  tea  cost,  at  $lT5r  a  pound? 
'25.  What  will  12J  lb.  .of  butter  cost,  at  12  cents 


.,    ,  ,    - 

'2C.  What  will  20  bushels  of  wheat  cost,  at  $£  a 
bushel  ? 

27.  What  will  7  chickens  cost,  at  $f  apiece? 

28.  What  will  9  turkeys  cost,  at  $1|  apiece? 

29.  What  will  12  geese  cost,  at  $J  apiece? 

30.  If  1  man  can  earn  1£  dollars  a  day,  how  much 
can  5  men  earn  ? 

31.  If  1  yard  of  cloth  cost  2|  dollars,  how  much 
will  2-|  yards  cost? 

32.  A  grocer    bought  J  of  a  box   of  raisins   for 
which  he  paid  2£  dollars  a  box  :  what  did  they  cost 
him  ? 

33.  If  f  of  a  sheep  is  worth  3  dollars,  what  will 
be  the  cost  of  2  sheep  ? 

34.  What  will  |  of  a  yard  of  cloth  cost,  at  7  dol 
lars  a  yard  ? 

35.  If  it  cost  2f  dollars  to  build  1  rod  of  wall,  how 
much  will  it  cost  to  build  3^-  rods  ? 

36.  James  gave  John  l^  apples  and  had  3  times  as 
many  left,  how  many  had  he  at  first? 

37.  If  a  yard  of  calico  cost  f  of  a  shilling,  what 
will  2J  yards  cost  ?  *• 

38.  What  will  12  yards  of  cloth  cost,  at  2|  dollars 
a  yard  ? 

39.  What  will  be  the  cost  of  6  turkeys  at  1J  dol 
lars  apiece  ? 

40.  If  2  yards  of  cloth  cost  8  J  dollars,  what  will 
12  yards  cost? 

41.  James  gave  8£  dimes  for  10  sheets  of  drawing* 
paper  :  how  much  was  that  a  sheet  ? 

42.  What  will  be  the  cost  of  a  bushel  of  wheat,  if 
8  bushels  cost  12  J  dollars  ? 


LB8.  XV.]  INTELLECTUAL    ARITHMETIC.  103 

LESSON    XY. 
Dividing  Fractions. 

1.  What  is  the  quotient  off  divided  by  21* 

2.  What  is  the  quotient  of  f  divided  by  2  ]  both 
ways. 

3.  What  is  the  quotient  of  f  divided  by  3  ?  both 
ways. 

4.  What  is  the  quotient  of  |  divided  by  3?  both 
ways. 

5.  What  is  the  quotient  of  l/  divided  by  5  ?  both 
ways. 

6.  What  is  the  quotient  of  *•£  divided  by  7  ?  both 
ways. 

7.  What  is  the  quotient  of  T82-  divided  by  4?  both 
ways. 

8.  What  is  the  quotient  of  -y8  divided  by  9?  both 
ways. 

9.  What  is  the  quotient  of  Tf  divided  by  6  ?  both 
ways. 

10.  What  is  the  quotient  of  f  divided  by  5? 

11.  What  is  the  quotient  of  ~  divided  by  3  ? 

12.  What  is  the  quotient  of  2g-4  divided  by  4  1 

13.  What  is  the  quotient  of  f  divided  by  9  ? 

14.  What  is  the  quotient  of  f  divided  by  2? 

15.  What  is  the  quotient. of  if  divided  by  8  1 

*  ANALYSIS. — Here  the  fractional  unit  is  5,  and  there  are 
6  taken.  Now,  if  we  divide  the  6  fractional  units  by  2,  the  quo 
tient  will  be  3  ;  that  is,  f . 

Again,  if  we  multiply  the  denominator  by  2,  we  shall  have 
j^,  in  which  the  fractional  unit  is  y-Q,  (which  is  the  half  of  J,) 
and  since  the  number  taken  is  the  same,  it  follows  that  y6(j  is  OE 
half  of  f .     Hence : 

A  fraction  may  be  divided  by  a  whole  number,  either  by 
dividing  the  numerator  or  multiplying  the  denominator  by  tht 
divisor. 


104  INTELLECTUAL    ARITHMETIC.  [SEO.  VI 

16.  What  is  the  quotient  of  T4¥  divided  by  4  ? 

17.  What  is  the  quotient  of  T3o-  divided  by  3  ? 

18.  What  is  the  quotient  of  j%  divided  by  3? 

19.  What  is  the  quotient  of  2^4  divided  by  12  ? 

20.  What  is  the  quotient  of  3£  divided  by  6  ? 

21.  What  is  the  quotient  of  5£  divided  by  7  1 

22.  What  is  the  quotient  of  9J-  divided  by  5? 

23.  What  is  the  quotient  of  12|  divided  by  8  ? 

24.  What  is  the  quotient  of  9|  divided  by  9  ? 

25.  What  is  the  quotient  of  5f  divided-by  7  ? 

26.  What  is  the  quotient  of  Sf  divided  by  43  ] 

27.  What  is  the  quotient  of  9-|  divided  by  7  ? 

28.  What  is  the  quotient  of  6|  divided  by  9? 

29.  What  is  the  quotient  of  1  ji  divided  by  23  ? 

30.  What  is  the  quotient  of  2T4^  divided  by  8  1 

31.  What  is  the  quotient  of  ^  divided  by  ft* 

32.  What  is  the  quotient  off  divided  by  f  ? 

33.  What  is  the  quotient  of  2|  divided  by  lrV? 

34.  What  is  the  quotient  of  C)^  divided  by  3f  1 

35.  What  is  the  quotient  of  8f  divided  by  3f  ? 

36.  What  is  the  quotient  of  3f  divided  by  2^3  ? 

37.  What  is  the  quotient  of  6-f-  divided  by  f  ? 

38.  What  is  the  quotient  of  5|  divided  by«7  ? 

39.  What  is  the  quotient  of  6^  divided  by  2^  ? 

40.  What  is  the  quotient  of  7-J-  divided  by  2T%  ? 

41.  What  is  the  quotient  of  8-f  divided  by  6-^  ? 

42.  What  is  the  quotient  of  5J  divided  by  5^-  ? 

43.  What  is  the  quotient  of  8^  divided  by  2£  1 

44.  What  is  the  quotient  of  3J-  divided  by  6f$  ? 

*  SUGGESTION. — Reduce  both  to  sixths  :  ^  is  equal  to  ^, 
and  -|  are  equal  to  ^. 

Now,  since  the  fractional  unit  is  the  same  in  both,  the  quotieui 
will  be  found  by  dividing  the  numerators :  Hence,  to  divide 
one  fraction  by  another, 

Reduce  them  to  the  same  fractional  unit,  and  then  divide  the 
numerator  of  the  dividend  by  the  numerator  of  tfie  divisor. 


LES.  XV. J  INTELLECTUAL    ARITHMETIC.  1 

45.  What  is  the  quotient  of  6f  divided  by  3J  1 

46.  What  is  the  quotient  of  sf  divided  by  2  ? 

47.  What  is  the  quotient  of  4f  divided  by 

48.  What  is  the  quotient  of  2|  divided  by 

49.  What  is  the  quotient  of  4|  divided  by  2^6 

50.  What  is  the  quotient  of  Sf  divided  by  6^  ? 

51.  What  is  the  quotient  of  3|  divided  by  2f  ? 

52.  What  is  the  quotient  of  6T3¥  divided  by  2|  ? 

53.  What  is  the  quotient  of  3|  divided  by  8|-  ? 
,     54.  What  is  the  quotient  of  9|  divided  by  3|  ? 

55.  How  many  times  is  f  contained  in  £?* 

56.  Plow  many  times  is   |  contained  in  2  ?     £  is 
contained  in  1    how  many  times'?     In  2  how  many 
times?     How  many  times  are  |  contained? 

57.  How  many  times  is  -|  contained  in  •§•  1 

58.  How  many  times  is  2£  contained  in  3  1 

59.  How  many  times  is  ^  contained  in  f  ? 

60.  How  many  times  is  -|  contained  in  2-J  1 

61.  How  many  times  is  |-  contained  in  |-  ? 

62.  How  many  times  is  2-J-  contained  in  |-  ? 

63.  How  many  times  is  ^  contained  in  -|  ? 

QUESTIONS. 

1.  If  3  yards  of  cloth  cost  $10^,  how  much  does  it 
cost  a  yard  ? 

2.  If  six  pounds  of  tea  cost  $4|,  what  does  it  cost 
a  pound  ? 

3.  If  John  gives  9y  cents  for  7  tops,  how  much  do 
they  cost  him  apiece  ? 

4.  If  7  pounds   of  sugar  cost  |    of  a  dollar,  how 
much  is  it  a  pound  1 

*  ANALYSIS. — One  is  contained  in  f ,  f  timfes.  But  £  is  con 
tained  in  -f,  5  times  as  many  times  as  1 ;  that  is,  l-f  times.  Bui 
2  fifths  is  contamed  half  as  many  times  as  •£- ;  that  is,  ^  times : 
Hence,  to  find  the  quotient  of  one  fraction  divided  by  another, 

Invert  the  terms  of  the  divisor  and  multiply. 
4 


10G  INTELLECTUAL    ARITHMETIC.  [SEC.  VI, 

5.  If  4  pounds  of  coffee  cost  $1 J,  how  much  does  it 
cost  a  pound  ? 

6.  If  7  oranges  cost  9J  cents,  how  much  do  they 
cost  apiece  ? 

7.  If  three  and   three  fourths  yards  of  cloth  cost 
§1 1 J,  how  much  does  it  cost  a  yard  ? 

8.  If  James  can  walk  14  miles  in  §  of  a  day.  hew 
far  can  he  walk  in  one  day  ? 

9.  If  John  can  buy  9  lemons  for  10J  cents,  how 
much  do  they  cost  him  apiece? 

10.  If  9   eggs  cost  IQj  cents,  how  much  do  they 
cost  apiece  ? 

11.  If  7|  bunches  of  grapes  are  worth  22-£  shillings, 
how  much  are  they  a  bunch  1 

12.  If  5£  bushels  of  potatoes  cost  $2f,  how  much 
do  they  cost  a  bushel  ? 

13.  If  nine  baskets  of  peaches  cost  $12-f ,  how  much 
are  they  a  basket? 

14.  If  8  lambs  cost  $12J-,  how  much  do  they  cost 
apiece  ? 

15.  If  9-f-  pounds  of  cheese  cost  $-|,  how  much  does 
it  cost  a  pound  ? 

1G.  If  4  barrels  of  flour  cost  24  dollars,  what  will 
Ijj  cost? 

17.  James  has   3   oranges   and   5  playmates:   he 
wishes  to  give  -i  of  an  orange  to  each,  \\ow  must  he 
divide  the  oranges,  and  how  many  oranges  will  he 
have  left? 

18.  If  one  man  consumes  1£  pounds  of  meat  in  a 
day,  how  many  men  would  8|  pounds  supply  ? 

19.  A  man  distributed  17|-  pounds  of  bvead  among 
8  persons,  how* much  does  he  give  to  each  ? 

20.  If  12  horses  consume  28|    tons  of  hay  in  a 
winter,  how  much  is  consumed  by  each  horse  ? 

21.  If  |  of  a  barrel  of  flour  will   last  a  family  30 
days,  how  long  will  2  barrels  last  them? 


LES.   XV.]  INTELLECTUAL    ARITHMETIC. 

22.  A  farmer  has  a  field  containing  8f  acres;  if  a 
man  can  mow  If  acres  in  a  day,  how  many  men  will 
it  take  to  mow  the  field  in  1  day  ? 

23.  If  9  bushels  of  apples  cost  4£  dollars,  how 
much  will  17  bushels  cost  ? 

24.  If  7  pounds  of  butter  cost  10-J-  shillings,  how 
much  is  that  a  pound  ? 

25.  If  3  pounds  of  butter  cost  77}  shillings,  how 
much  will  12  pounds  cost? 

26.  If  5  yards  of  cloth  cost  11 J  dollars,  what  will 
8  yards  cost  ? 

*27.  If  8    yards  of  cloth  cost  42-£  dollars,  how  much 
will  4  yards  cost? 

28.  If  If  dollars  will  buy  2  yards  of  cloth,  how 
many  yards  will  6  dollars  buy  1 

29.  How  manf  times  is  2J  contained  in  124-  ? 

30.  How  many  pounds  of  tea  can  be  purchased  for 
6|-  dollars,  if  it  cost  -J-  dollars  a  pound  ? 

31.  If  a  turkey  costs  If  dollars,  how  many  can  be 
bought  for  12|  dollars? 

82.  If  calico  is  worth  £  of  a  dollar  a  yard,  and 
muslin  }  of  a  dollar  a  yard,  how  much  calico  must  be 
given  for  9  yards  of  muslin  ?  What  is  the  cost  of  9 
yards' of  muslin  ? 

33.  A  tailor  bought  a  piece  of  cloth  containing  12| 
yards,  for  which  he  paid 38  dollars  :  what  did  it  cost 
him  a  yard? 

34.  If  you  give  15|  dollars  for  a  cow,  and  sell  her 
for  \  more  than  she  cost,  how  much  more   do  you 
receive  for  her  than  you  gave  ? 

35.  If  6  men  can  do  a  piece  of  work  in  15|  clays. 
how  long  will  it  take  1  man  to  do  it?     2  men?     3 
men  ? 

36.  A  man  divided  2~  of  a  dollar  among  his  children, 
giving  j7^  of  a  dollar  to   each  :   how  many  children 
were  there. 


INTELLECTUAL    ARITHMETIC.  [SEC.  VIL 

SECTION    SEVENTH. 


LESSON   I. 
Comparison  of  Numbers. 

} .  What  part  of  2  is  1  1     How  many  times  does  1 
on  tain  2?* 

2.  What  part  of  3  is  2?     What  is  the  ratio  of 
to  21 

3.  What  part  of  6  is  3?     How  many  times  does 
8  contain  6  ? 

4.  What  part  of  12  is  4?     What  is  the  ratio  of 
12  to  4  1 

5.  What  part  of  4  is  1  ?     What  is  the  ratio  1 

6.  What  part  of  3  is  2  ?     What*s  the  ratio  ? 

7.  What  part  of  8  is  5  1     What  is  the  ratio  ? 

8.  What  part  of  20  is  10?     What  is  the  ratio  ? 

9.  What  part  of  30  is  20  ?     What  is  to  ratio  ? 

10.  What  part  of  50  is  10  ?     What  is  the  ratio? 

11.  What  part  of  £  is  i  ]     What  is  the  ratio  1 

12.  What  part  of  j  is  f  ?      What  is  the  ratio  ? 

13.  What  part  of  f  is  f-  ?     What  is  the  ratio  ? 


*  SUGGESTIONS. — To  find  what  part  any  number  is  of  a  num 
ber  less  than  itself,  we  divide  the  less  by  the  greater,  and  the 
quotient  shows  the  part.  This  quotient  is  called  the  ratio. 
Thus,  what  part  of  2  is  1  ;  it  is  1  divided  by  2,  which  is  |. 

If  we  wish  to  find  how  many  times  one  number  is  greater 
than  another,  we  divide  the  greater  by  the  less,  and  the  quotient 
shows  the  number  of  times.  This  quotient  is  also  called  the 
ratio  of  the  two  numbers.  Thus,  how  many  times  is  12  greater 
than  2:  it  is  12  divided  by  2  times  greater,  which  is  6  times. 

But  the  general  question,  "  How  many  times,"  has  no  refer 
ence  to  the  relative  value  of  the  numbers.  Thus :  what  part 
of  2  is  1  ?  Or,  how  many  times  is  2  contained  in  1  \  One  hall; 
or  one  half  times?  How  many  times  is  2  contained  in  41 
Twice,  or  2  times. 


LEfc.    I.]  •  INTELLECTUAL    ARITHMETIC.  10D 

14.  What  part  of  |  is   |  1     What  is  the  ratio  ? 

15.  What  part  of  |  is  T\  ?     What  is  the  ratio  '? 

16.  What  part  of  |  is    f  ?     What  is  the  ratio  1 

17.  What  part  of  3i  is  3f  ?     What  is  the  ratio  ? 

18.  What  part  of  4|  is  3J  ?     What  is  the  ratio  ? 

19.  What  part  of  3±  is  2f  ?     What  is  the  ratio  ? 

20.  What  part  of  8f  is  5£  ?     What  is  the  ratio  ? 

21.  What  part  of  8  is  f  of  2-J-?     What   is    the 
ratio? 

22.  What  part  of  10  is  f  of  4|  ?     What  is  the 
ratio  ? 

23.  What  part  of  15  is  4  ? 

24.  What  part  of  45  is  12? 

25.  What  part  of  |  is  -§-  ? 

20.  What  is  the  ratio  of  12  to  7  7 

27.  What  is  the  ratio  of  H  to  i  • 

28.  What  is  the  ratio  of  3f  to  1T^  ? 

29.  What  is  the  ratio  of  9  to  f  of  7|  1 
' 


.          a     s      e  rao  o         o      o 
30.    What  part  of  7£  is  1£  of  $£  1 


QUESTIONS. 

1.  Vv  i.at  part  of  30  dollars  is  5  dollars  ?     6  dollars? 
1J  dollar?     2  dollars? 

2.  Wiwc   part    of   56    bushels    is   8   bushels?     7 
bushels  ?     4  bushels  ?     28  bushels  ? 

i>.  What  part  25  yards  is  -|  of  16  yards? 

4.  A   tailor  has   12  yards  of  cloth  ;  he  cuts  off  o 
yardb  for  a  coat,  1^  yards  for  a  pair  of  pantaloons, 
1-|  yards   for   a   vest  :    what  part  of  tjhe  cloth  did  lie 
use  ?      What  part  was  left  ? 

5.  A  farmer  has  60  bushels  of  wheat  in  his  barn  ; 
e   .sells  20  bushels  to  one  man,  15  to  another,  ami 

iO  to  another  :  what  part  of  the  wheat  has  he  left  ? 

6.  Mr.  Dix  buys  f  of  £  of  f  of  Mr.  Jones'  farm, 
hich  contains  100  acres:   what  part  of  the  farm  does 

buy  ?     How  many  acres  are  there  left  2 
10 


110  INTELLECTUAL    ARITHMETIC.    *        [SEC.  VII. 

7.  James  gave  5  dollars  for  his  hat,  and  his  father 
gave  25  dollars  for  a  coat:  what  part  of  the  cost  of 
the  coat  was  the  cost  of  the  hat  1 

8.  What  part  of  a  barrel  of  flonr  is  £  of  f  of  it  ? 
What  part  of  the  whole  barrel    is  the    remainder? 
What  will  each  part  cost  if  flour  is  8  dollars  a  barrel  ? 

9.  William    having    a    pine-apple    said    he    would 
give  |  of  f  of  it  to  the  one  that  could  tell  how  much 
that  would  be  :  how  much  did  the  individual  receive  ? 

'  What  part  of  the  pine-apple  was  left? 

10.  A  merchant  ha<3  4j  barrels  of  flour  and  sold  | 
of  it :  how  much  had  he  left  1 

11.  A  lady  paid  f  of  51  dollars  for  a  pencil  and 
1-|  dollars  for  a  ring  :   how  many  times  as  much  did 
she  pay  for  the  pencil  ns  for  the  ring  ? 

12.  One   man  can   build   5|-  rods   of  wall' in   one- 
day,  and  another  can  build  3^  rods:  what  part  is  the 
second  of  the  first  1 

13.  Albert  is  9y  years  of  age,  John  is  |-  of  Albert's 
age  :  how  old  is  John  1  % 

14.  James  bought  a  kite  for  16 1-  cents,  which  was 
t>i  times  as  much  as  he  paid  for  his  top  :  how  much 
did  he  pay  for  his  top  1 

15.  A   fanner  bought  a  calf  for  4£  dollars,  and  a 
pig  for  1J  of  that  sum  :  what  did  he  pay  for  the  pig  ? 

16.  A  box  of  raisins  cost  3J  dollars,  at  tne  rate  of 
i  of  a  dollar  a  pound:  how  many  pounds  were  there 
in  the  box  1 

17.  If   14  oranges  are  worth  as  much  as  3£  pine 
apples,  how   many  oranges  is  one  pine-apple  worth'? 

18.  Mary  gave  |  of  3^  dollars  for  a  bonnet,  which 
was  %  of  what  Jane  paid  for  hers:   how   much   did 
Jane  pay  ? 

19.  If  a  father  can  do  1|  times  as  much  work  in  a 
day   as    his    son,  how  many  days'  work  of  the  father 
will  be  equal  to  18  of  the  son's? 


LES.  II.]  INTELLECTUAL    ARITHMETIC.  Ill 

20.  A  gentleman  gave  200  dollars  to  his  three 
sons  ;  to  the  first  he  gave  J  of  it,  to  the  second  |  of 
2  times  as  much,  and  the  rest  to  the  third  :  how 
much  had  each '? 

21..  If  a  boy  can  run  80  rods  in  eight  minutes, 
what  part  of  a  mile  can  he  rim  in  16  minutes  ] 

22.  James  bought  -J   pound  of  candy,  and  gave  | 
of  it  to  Mary  :   what  part  of  a  pound  did  he  give  her  'I 

23.  A    vessel    sails    150    miles    one    day,    and    is 
retarded  \  of  £  as  much  on  the  second  :  what  part 
of  150  miles  did  she  sail  the  second  day  1 


LESSON    II. 
Comparison  of  Numbers. 

1.  Two  is  one  half  of  what  number?* 

2.  Four  is  one  third  of  what  number? 

3.  Five  is  one  fifth  of  what  number] 

4.  Two  and  one  half  is  one-fourth  of  what  number  ? 

5.  One  and  one  half  is  one  sixth  of  what  number  '* 

6.  One  and  2  thirds  is  one-third  of  what  number  ?f 

7.  Three  and  one  third  is    one    third    of    what 
number  1 

8.  Ten  and  five  sixth  is   one    sixth   of  what  num 
ber  ? 

0.  Twelve  is  one  fifth  of  what  number  ? 

10.  Eight  and  three  fourths  is  one  fourth  of  what 
r,  timber  ? 

11.  Nine  and  one  sixth  is  one  sixth  of  what  num 
ber  ? 

12.  Five  and  one  tenth  is  one  tenth  of  what  num 
ber  '? 


*  ANALYSIS. — Two   is  one  half  of  2  times    2,  which  aro  4 
therefore,  2  is  one  half  of  4. 

f  ANALYSIS. — One  and  2  thirds  is  one  third  of  3  times  one  ami 
2  thirds,  which  are  5  :  therefore,  one  and  two  thirds  is  one  third 
of  5. 


11*2  INTELLECTUAL    ARITHMETIC.  [SEC.  Vlt, 

13.  Five  and  f  is  one  fifth   of  what  number  ] 

14.  iSix  and  three  fourths  is  one  eighth  of  what 
number  1 

15.  Twelve  and    one  sixth  is  one  sixth  of  -what 
nmnoer  1 

16.  Two  and  three  fourths  is  one  twelfth  of  what 
number  1 

17.  Three   and    six  ninths   is  one  ninth   cf  what 
number] 

18.  Six  and  %  is  one  twelfth  of  what  number? 

19.  Eight  is  2  thirds  of  what  number]* 

20.  12  is  3  fourths  of  what  number] 

21.  16  is  4  fifths  of  what  number  ] 

22.  30  is  6  sevenths  of  what  number] 

23.  32  is  8  ninths  of  what  number] 

24.  15  is  5  ninths  of  what  number] 

25.  12  is  4  sevenths  of  what  number  ] 

26.  22  is  11  twelfths  of  what  number? 

27.  27  is  9  tenths  of  what  number  ] 

28.  21  is  7  ninths  of  what  number  ] 

29.  15  is  3  sevenths  of  what  number] 

30.  24  is  8  elevenths  of  what  number  ] 

31.  16  is  5  times  what  number  ]j- 

32.  48  is  6  times  what  number  ] 

33.  39  is  3  times  what  number  ] 

34.  56  is  9  times  what  number  ? 

35.  21  is  5  times  what  number] 

36.  75  is  8  times  what  number] 

37.  54  is  8  times  what  number] 

38.  95  is  9  times  what  number  ? 

*  ANALYSIS. — Since  8  is  2  thirds  of  some  number,  one  half  <,f 
8,  Tvhicli  is  4,  is  one  third  of  the  same  number ;  but  4  is  one 
third  of  3  times  4,  which  are  12 ;  therefore,  8  is  2  thirds  of  12. 

f  ANALYSIS. — 16  is  5  times  -J-  of  16,  which  is  83-:  therefore,  Itf 
is  6  times  34. 


LES.  II.]  INTELLECTUAL    ARITHMETIC.  113 

QUESTIONS. 

1.  Two  thirds  of  nine  is  one  half  of  what  number? 

2.  Two  se\enths  of  fourteen  is  one  third  of  what 
number  ? 

3.  Three  twelfths  of  thirty-six  is  one  eighth  of  what 
number? 

4.  James  gave   nine   cents  for  a  slate,  which  was 
three  fourths  of  his  money:  how  much  had  he? 

5.  A  man  bought  a  cow,  for  which  he  paid   $30. 
which  was    three    fifths    of  his    money :  how    much 
had  he? 

6.  A  lady  was  married  at  twenty  years   of  age, 
which  was  the  half  of  eight  fifths  of  the   age  of  her 
husband  :  how  old  was  the  husband  ? 

7.  In  a  pasture  are  45  sheep,  which  is  three  fourth 
the  number  of  cows  in  the  same  pasture  :  how  many 
cows  are  there? 

8.  John  gave  36  cents  for  a  knife,  which  was  six 
sevenths  of  what  he  gave  for  a  sled  :  how  much  did 
he  give  for  his  sled  ? 

9.  If  a  man  can  make  six  and  two  ninths  rods  ot 
fence  in  one  day,  how  much  can  he  make  in  12  days? 

10.  Two   men  agreed  to  build   a  wall  ;  one   man 
built  16- rods,  which  was  four  fifths  of  what  the  other 
built :  how  much  did  the  second  build,  and  what  was 
the  whole  length  of  the  wall? 

11.  A  man  willed  one  half  of  his  property  to  his 
wife,  one  third  of  the  remainder  to  his  daughter  Mary, 
$nd  one  sixth  to  his  son  James:  how  much  of  it  was 
left  for  his  only  remaining  son  John  ? 

12.  John  gave  one  and  3  fourth  cents  for  a  peach 
which  was  one  third  of  what  he  gave  for  an  orange 
what  did  the  orange  cost  him  ? 

13.  Charles  gave  ten  and  five  sixths  cents  for  his 
kite,  which  was  five  times  what  he  paid  for  his  top : 
how  much  did  he  pay  for  his  top?     How  much  for 
both?  10* 


114  INTELLECTUAL    ARITHMETIC.  [SEC.  VII. 

14.  William  gave  eijjht  and  five  ninths  cents  for  a 
pencil,  which  was  one  fourth  the  cost  of  his  penknife : 
what  did  his  penknifa  cnst  him'? 

15.  A  farmer  paid  four  and  three  fourths  dollars 
for  a  calf,  and  one  fifth  as  much  for  a  pig:  what  did 
the  pig  cost  him  ? 

16.  A  farmer  bought  a  calf  for  three  and  one  sev 
enth  dollars,  which  was  one  seventh  of  what  he  paid 
for  a  cow  :  what  did  the  cow  cost  him  ? 

17  A  man  bought  a  pair  of  boots  for  six  and  £ 
dollars,  and  a  coat  which  cost  him  three  and  a  half 
times  as  much  :  what  did  the  coat  cost  him  ? 

18.  If  3  sevenths  of  a  barrel  of  flour  cost  G  dollars, 
what   will  5  sevenths  cost?     What  will  the  whole 
barrel  cost  ? 

19.  A  grocer  bought  5  boxes  of  raisins   for   two 
and  three  fifths  dollars  a  box,  and  a  barrel  of  sugar, 
which  cost  one  half  of  two  thirds  as  much  as  the  rai 
sins  :  how  much    less    did  the  raisins  cost  him  than 
the  sugar? 


LESSON   III. 
Comparison  of  Numbers. 

1.  25  is  5  eighths  of  how  many  times  7  ?* 

2.  63  is  7  ninths  of  how  many  times  8  ? 

3.  36  is  4  sevenths  of  how  many  times  6  ? 

4.  45  is  5  sixths  of  how  many  times  5? 

5.  84  is  7  eighths  of  how  many  times  9  ? 

6.  29  is  3  ninths   of  how  many  times  10? 

7.  42  is  7  thirds  of  how  many  times  5? 

8.  64  is  8  fifths  of  how  many  times  3  ? 

9.  32  is  3  eighths  of  how  many  times  11? 

*  ANALYSIS. — If  25  is  5  eig-hths  of  some  number,  1  eighth  is 
1  fifth  of  25,  which  is  5.  If  5  is  1  eighth,  8  eighths  are  8  times 
5,  which  are  40.  As  many  as  7  is  contained  times  in  40  :  7  is 
contained  in  40,  5  and  5  sevenths  times;  therefore,  25  is  5 
eighths  of  6  times  7  and  5  sevenths  of  7 


LES.  III.]  INTELLECTUAL    ARITHMETIC.  115 

10.  75  is  5  eighths  of  how  many  timbs  6  ? 

11.  21  is  7  fifths  of  how  many  times  4.  'I 

12.  40  is  4  sevenths  of  how  many  times  3 1 

13.  6  sevenths  of  21  is  3  fifths  of  what  number  ?* 

14.  8  ninths  of  27  is  4  sevenths  of  what  number'? 

15.  3  fourths  of  40  is  6  tenths  of  what  number  ? 

16.  5  eighths  of  64  is  8  ninths  of  what  number  1 

17.  2  thirds  of  42  is  5  twelfths  of  what  nurn  ber  ? 

18.  7  tenths  of  80  is  8  ninths  of  what  number? 

19.  4  fifths  of  60  is  6  sevenths  of  what  number? 

20.  5  sixths  of  48  is  7  eighths  of  what  number? 

21.  3  sevenths  of  21  is  2  thirds  of  what  number? 

22.  4  ninths  of  81  is  5  ejghths  of  what  number? 

23.  7  fifths  of  45  is  9  sevenths  of  what  number? 

24.  5  ninths  of  36  is  4  fifths  of  how  many  times  5  ? 

25.  4  sevenths  of  56  is  8    ninths    of   how  many 
fimes  7  ? 

26.  3   fourths  of  24  is  6  sevenths  of  how  many 
times  4  ? 

27.  5  eighths  of  48  is  5  ninths  of  how  many  times  9  ? 

28.  4  thirds  of  30  is  8  elevenths  of  how  many 
times  8 1 

29.  5  tenths  of  72  is  4  ninths  of  how  many  times  6  ? 

30.  11   twelfths  of  84  is  7  ninths  of  how   many 
times  10? 

31.  6  fifths  of  50  is  4  thirds  of  how  many  times  12  ? 

*  ANALYSIS. — 6  sevenths  of  21  is  6  times  1  seventh  of  21 :  1 
seventh  of  21  is  3,  and  6  sevenths  is  6  times  3,  which  are  !£, 
If  18  is  3  fifths  of  some  number,  1  third  of  18  which  is  6  is  1 
fifth  of  the  same  number ;  6  is  1  fifth  of  5  times  6  which  is  3D , 
therefore,  6  sevenths  of  21  is  3  fifths  of  30. 

VERIFICATION.— One  fifth  of  30  is  6,  and  3  fifths  of  30  are  18 
out  6  sevenths  of  21  are  18  :  therefore,  30  is  the  number  sought, 

NOTE. — With  the  correct  answer  all  the  questions  may  be 
reversed  and  similarly  analyzed :  thus,  30  is  5  thirds  of  6 
sevenths  of  what  number  ?  Again,  30  is  6  thirds  of  how  many 
sevenths  of  21  ? 


110  INTELLECTUAL    ARITHMETIC.  [SEC.  VIL 

32.  9   elevenths  of  88  is   8  fifths  of  how   many 
times  6? 

33.  2  thirds  of  75  is  5  sixths  of  how  many  times  9  ] 

34.  5  halves  of  24  is  3  fourths  of  how  many  times  5  \ 

35.  5  ninths  of  72  is  4  thirds  of  what  number  ? 

36.  6  sevenths  of  56  is  5  ninths  of  what  numbei  I 

37.  28  is  1  fifth  of  how  many  times  9  1 

38.  42    is     7     twelfths  of  how  many  times  12  ? 

39.  5  ninths  of  108  is  10  elevenths  of  how  many 
times  10? 

40.  6  fifths  of  35  is  7  twelfths  of  how  many  times  5  ? 

41.  81  is  9  fourths  of  how  many  times  8? 

42.  7  thirds  of  36  is  9  tenths  of  what  number] 

43.  7  twelfths  of  96  is  8   ninths    of  how   many 
times  5? 

QUESTIONS. 

1.  A  boy  gave  5  apples  to  one  of  his  companions, 
which  was  one  third  of  all  he  had  :   how  many  had  he? 

2.  A  man  bought  a  watch  for  24  dollars,  and  sold 
it  for  four  fifths  of  what  it  cost  him:  what  did   he 
receive    for    it,   and  how  much  did  he  lose  by  the 
bargain  ? 

3.  Charles  gave  away  12  apples  to  3  of  his  com 
panions,  which  was  |  of  f  of  all  he  had  :  how  many 
had  ne,  and  how  many  did  he  give  each? 

4.  If  3  fourths  of  a  hundred  weight  of  sugar  cost  12 
dollars,  what   will   a   hundred  weight   cost  ?     How 
many  barrels  of  cider  at  2  dollars  a  barrel  will  pay 
lor  it  ? 

5.  A  man  sold  a  horse  for  60  dollars,  which  was  5 
sevenths  of  what  he  cost  him  :  how  much  did  he  cost 
him,  and  how  much  did  he  lose?     When   he  bought 
him  he  paid  in  cloth  at  6  dollars  a  yard :  how  many- 
yards  of  cloth  did  he  give? 

6.  A  pole  is  |-  in  the  water,  ^  in  the  mud,  and   14 
feet  out  of  the  water  :  how  long  is  the  pole  ? 


LES.  III.]  INTELLECTUAL    ARITHMETIC.  11? 

7.  John  s  age  is  |  William's,  and  the  sum  of  their 
ages  diminished  by  5  is  equal  to  70:  what  is  the  age 
of  each  ? 

8.  Mr.  Wilson  gave  200  dollars  to  his  neice,  which 
was  J-  of  |-  of  his  property,  and  the  remainder  equally 
to  his  4  sons  :  how  much  did  each  receive  ? 

9.  Plow  many  yards  jof  cloth,  at  4  dollars  a  yard, 
must  be  given  for  a  hogshead  of  sugar,  if  four  sev 
enths  of  it  cost  48  dollars  ? 

10.  There  is  a  pole  standing  ^  in   the  water,  l  of 
the  remainder  in  the  mud,  and  6  feet  above  the  water  : 
how  long  is  the  pole  ? 

11.  A  staff  5  feet  long  casts  a  shadow  of  3  feet  : 
what  is  the  length  of  a  pole  that  casts  a  shadow  of  24 
feet  the  same  time  of  day  ?* 

12.  A  man  can  build  56  rods  of  wall  in  a  certain 
time,  another  man  can  build  8  rods  while  the  first 
builds  7  :   how  much  would  the  second  build  in   the 
same  time? 

13.  Two  boys  counting  their  marbles,  one  said  he 
had  sixteen.     The  other  said,  3  eighths  of  yours  is 
exactly  2  sevenths  of  mine:   now  if  you  will  tell   me 
how   many   I  have,   I   will    give   you   the   difference 
between  yours  and  mine:   how  many  had  he? 

14.  A  man  being  asked  how  many  sheep  he  had, 
said  he  had  them  in  three  pastures:  in   the  first  he 
nad  42,  which   was  7  eighths  of  what  he  had  in  the 
second  ;  and  that  5  thirds  of  what  he  had  in  the  second 
was  just  4  times  what  he  had  in  the  third  :  how  many 
had  he  in  each  field  ? 

15.  A  gambler  lost  3  fourths  of  his  money  in  play; 
he  then  won  30  dollars,  which  was  5«ixths  of  what  lie 
lost  :   how  much  money  had  he  when  he  began  to 


*  ANALYSIS.  —  Since  the  shadow  of  the  staff  is  3  fifths  the 
length  of  the  staff,  the  shadow  of  the  pole  must  be  3  fifths  the 
length  of  the  pole  ;  then  if  24  is  3  fifths  of  some  number  tfco, 


118 


INTELLECTUAL    ARITHMETIC.  [SEC. 


16.  A  tailor  cut  off  3  fifths  of  a  piece  of  cloth,  he 
then  cut  off  4  yards,   which  was  one   third  of  the 
remainder  :  how  many  yards  were  there  in  the  piece? 

17.  A  gentleman  left  to  his  eldest  son  300  dollars, 
which  was  3  fourths  of  what  the  second  son  had,  and 
twice  the  second  son's  share  was  just  four  times  what 
the  third  son   received  :    how  much  was   the    fathe 
worth  1 

18.  James  being  asked  how  many  credit  marks  he 
had,  said  :   if   1  third  of  the  number  be  taken   from  1 
half  of  the  number,  the  remainder  would  be  2£  tiroes 
4:   how  many  credits  had  he1? 

19.  Three   fourths  of  40  is    5  sevenths   as  many 
dollars  as  Mr.  C.  paid  for   his  horse:  what  was  the 
cost  of  the  horse  ? 

20.  A   person  being    asked   his    age    said,  that  3 
fourths  of  80  was  6  sevenths  of  5  times  his  age  :  what 
was  his  age  ? 

21.  Bought  45  yards  of  cloth  and  sold  4  ninths 
of  it   for  20  dollars,  which  was  5  sixths  of  what  the 
whole  cost :  what  would  be  the  gain  on  the  whole,  afc 
the  same  rate  1 

22.  A  merchant   sold    6    barrels    of    flour  for  54 
dollars,  which   was  9  eignths  of  what  it  cost  him  : 
what  did  it  cost  him  a  barrel? 


LESSON    IV. 

1.  3  sevenths  of  56  is  8  ninths  ^of  3  times  what 
number  ?* 

*  ANALYSIS. — Three  sevenths  of  56  is  3  times  1  seventh  of  56. 
One  seventh  of  56  ^s  8,  and  3  sevenths  of  56  is  3  times  8  which 
ar3  24.  Since  24  is  8  ninths  of  some  number,  1  eighth  of  24 
which  is  3  is  1  ninth  of  ^he  same  number:  3  is  1  ninth  of  9 
times  3  which  is  27.  Now,  27  is  3  times  1  third  of  27  which 
\B  9 ;  therefore,  3  sevenths  of  56  is  8  ninths  of  3  times  y, 
or  27. 


LES.  IV.]  INTELLECTUAL    ARITHMETIC.  119 

2.  5  sixths  of  54  is  5  eighths  of  nine  times  \vhat 
number'? 

3.  4  fifths  of  30   is   3  fourths   of  8   times  what 
number  ? 

4.  2  ninths  of  81  is  3  elevenths  of  4  times  what 
number  ? 

5.  5  eighths  of  56  is  7  ninths  of  6  times  what 
number  ? 

6.  4  thirds  of  36  is   6  fifths   of  10   times  what 
number  1 

7.  9  tenths  of  90  is  6  fourths  of  8  times  what 
number  ? 

8.  6  halves  of  30  is  9  tenths  of  20  times  what 
number  1 

9.  7  ninths  of  108  is  7  twelfths  of  8  times  what 
number? 

10.  5  eighths  of  32  is  4  fifths  of  how  many  sixths 
of  18?* 

11.  6  sevenths  of  56   is  8  ninths  of  how   many 
fourths  of  24  ? 

12.  8  fifths  of  40  is  4  thirds  of  how  many  sixths 
of  36? 

13.  4  thirds  of  36  is  8  twelfths  of  how  many  fifths 
of  45? 

14.  2  fifths  of  75  is  5  sevenths  of  how  many  ninths 
of  54? 

15.  3  halves  of  40  is  6  twelfths  of  how  many 
tenths  of  80  ? 

16.  7  ninths  of  72  is  8  fifths  of  how  many  thirds 
of  24? 

17.  6  elevenths  of  44  is  3  tenths  of  how  mai.y 
fourths  of  32? 

*  ANALYSIS. — The  same  as  in  the  preceding  examples  until 
you  obtain  the  second  number,  which  in  this  example  is  25. 
Then,  25  is  how  many  sixths  of  18  ?  1  sixth  of  18  is  3,  and  3  is 
contained  in  25  8  and  1  third  times ;  therefore,  5  eighths  of  32 
is  4  fifths  of  8  and  1  third  times  1  sixth  of  18. 


120  INTELLECTUAL    ARITHMETIC.  [SEC.  VII 

18.  5  sevenths  of  77  is  11  twelfths  of  how  manv 
eighths  of  56  ? 

19.  10  thirds  of  24   is   8  halves   of  how   many 
twelfths  of  36? 

20.  12  fifths  of  45  is  9  tenths  of  how  many  eighths 
of  64? 

21.  5  sixths  of  54  is  3  fourths  of  how  many  sev 
enths  of  42  ? 

22.  4  fifths  of  30  is  4  sevenths  of  how  many  times 
2  thirds  of  21  1* 

23.  6  thirds  of  27  is  6  ninths  of  how  many  times 

1  tenth  of  90  ? 

24.  8  ninths  of  63  is  7  twelfths  of  how  many  times 
4  eighths  of  24? 

25.  7  sixths  of  54  is  9  tenths  of  how  many  times 

2  ninths  of  45  ? 

26.  5  halves  of  24  is  5  eighths  of  how  many  times 

4  sevenths  of  28  '? 

27.  9  tenths  of  70  is  7  sixths  of  how  many  times 

3  ninths  of  27? 

28.  4  thirds  of  36  is  4  ninths  of  how  many  times 
2  sevenths  of  42  ? 

29.  5  ninths  of  72  is  4  fifths  of  how  many  times 

5  twelfths  of  60? 

30.  7  eighths  of  64  is   8  sevenths  of  how  many 
times  3  eighths  of  32? 

31.  10  fourths  of  36  is  9  thirds  of  how  manv  sev 
enths  of  63  ? 


*  NOTE — In  connection  with  the  answer,  a  reversed  state 
went  of  the  examples  in  this  lesson  may  be  made,  giving  two 
other  propositions,  to  be  solved  by  a  similar  analysis.  Thus,  in 
Example  22,  3  being  the  answer  we  have, 

First.  3  times  2  thirds  of  21  is  7  fourths  of  4  fifths  of  what 
number  ?  which  will  give  30. 

Second.  3  times  2  thirds  of  21  is  7  fourths  of  how  many  fifths 
of  80  ?  which  will  give  ±  fifths  of  80. 

This  will  give  the  pupil'the  benefit  of  three  examples  in  cue 


LES.  III.]  INTELLECTUAL    ARITHMETIC.  121 

32  G   sevenths  of  77  is  11  twelfths  of  how  inanv 
fifths  of  50  ? 

33.  8  ninths   of  81  is  6  fifths    of  4  times  what 
number  ? 

34.  9  tenths  of  100  is  5  halves  of  8  times  what 
number  ? 

35.  3  sevenths  of  84  is  4  ninths  of  how  many  times 
4  ninths  of  45. 

QUESTIONS. 

1.  Two  boys  comparing  their  ages,  one  said  he  was 
fifteen   years  old;  the  other  said,  4  fifths  of  your  age 
is  just  3  halves  of  my  age  :  what  was  his  age  ? 

2.  A  farmer  had  a  certain  number  of  sheep  which 
he  put  in  two  fields;   in    one  field   he  had  28,  and  G 
sevenths  of  them  was  4  ninths  of  2  times  what  he  had 
in   the   second:  how   many  were  there  in  the  second 
field  f 

3.  A  man  pays   300  dollars  a  year  for  benevolent 
objects:  -|  of  this  sum  is   equal  to  y  of  2  times  the 
amount  of  his  personal  expenses  :  what  are  his  persona] 
expenses  ? 

4.  A  fanner  sold  a  number  of  cows  and  had  12  left, 
which  was  J  of  the  number  sold  ;  if  the  number  sold 
be  divided  by  f-  of  9j,  the  quotient  will  be  -J-  the  num 
ber  of  dollars  he  received  per  head :  how  much  did  IK 
receive  apiece  for  his  cows  ? 

5.  The  insurance  on  a  house  is  600  dollars,  and  $  of 
that  is  f  of  4  times  the  value  of  the  furniture:  what 
is  the  furniture  worth  1 

G.  A  man  bought  a  horse  for  100  dollars,  |-  of  what 
the  horse  cost  .was  -|  of  what  he  paid  for  a  carriage- 
how  much  did  the  carriage  cost  him? 

7.  y  of  A's  age  is  -|  ot  B's,  and  3  times  B"s  is  |  of 
C's :  how  old  are  A  and  B,  if  C  is  24  years  old  ? 

8    Fort  Plain  is  5G  miles  from   Albany,  y  of  this 
distance  is   |    times  J-  the  distance  from  Albany  to 
Rochester  :  what  is  the  distance  to  Rochester? 
11 


122  INTELLECTUAL    ARITHMETIC.  [SEC.    VII. 

9.  The  contents  of  a  certain  store  cost  1,000  dol 
lars,  and   f  the  entire  cost  is  |  of  3  times  what   the 
cloths  cost :  what  was  the  cost  of  the  cloths  ? 

10.  James  has  a  certain  number  of  marbles;  John 
has  |  as  many  less  3,  and  William  has  ~  as  many  as 
*olm  less  7  ;  William  has  5  marbles :  how  many  have 

ohn  and  James? 

11.  A  house  is  worth  600   dollars,   and   |-  of  its 
value  is  ^  of  2J  times  the  value  of  the  farm  on  which 
it  stands  :  what  is  the  value  of  the  farm  ? 

12.  Buffalo  is  3*25  miles  from  Albany,  and  |  of  this 
distance  is  3|  times  J  the  distance  from  Rochester  to 
Buffalo  :   what  is  the  distance  1 

13.  A  boy  being  asked  his  age  said,  that  9  years 
was  3  years  more  than  f  times  |-  of  his  age  :  what  was 
his  age  1 

14.  A  man  had  |-  of  his  money  stolen  from  him  ; 
the  thief  was  caught,  but  not  until  he  had  spent  f  of 
it,  the  remainder,  (850),  was  given  back:  how  much 
money  had  he  at  first  ? 

15.  A  and  B  engaged   in   play  with  equal  sums  of 
money,  B  gained  40  dollars,  which  was  |  of  3  times 
what  he  commenced  with  :  how  much  had  each  when 
they  began  to  play  ?     How  much  had  A  left  1 

15.  A  farmer  sold  a  horse  fur  96  dollars,  which 
was  |  times  |  what  he  paid  for  him:  how  much  di<J 
he  pay  for  him  ] 


LESSON    V. 

Comparison  of  the  units  of  Denominate  Numbers, 

1.  Four  mills  are  what  part  of  a  cent  ? 

2.  Five  cents  are  what  part  of  a  dollar? 

3.  Three  dimes  are  what  part  of  a  dollar  ? 

4.  Thirty -six  cents  are  what  part  of  a  dollar  1 


LKS.    V.]  INTELLECTUAL    ARITHMETIC.  123 

5.  Three  dollars  is  what  part  of  an  eagle? 

6.  Plow  many  cents  in  |  of  a  dollar? 

7.  How  many  dimes  in  f  of  a  dollar? 

8.  How  many  mills  in  T30  of  a  dollar? 

9.  How  many  dollars  in  |  of  an  eagle  1 

10.  What  part  of  a  pound  is  1  shilling'? 

11.  'What  part  of  a  pound  is  6  pence  ? 

12.  What  part  of  a  shilling  is  5  pence? 

13.  What  part  of  a  shilling  is  3  farthings  1 

14.  What  is  the  value,  in  pence,  of  f-  of  a  shilling'! 
Of  f  of  a  shilling  ?     Of  £  of  a  pound  ? 

15.  What  is  the  value,  in  shillings  and  pence,  of  f 
of  £  of  a  pound  ?     Of  ^  of  f  of  \  of  a  pound  ? 

16.  Seven  pence  is  what  part  of  a  pound  ? 

17.  Eleven  pence  is  what  part  of  a  shilling  ?  Of  £  1 

18.  What  part  of  a  pound  is  6  shillings  ?     7  shil 
lings?     12   shillings?     13    shillings?     14   shillings? 
15  shillings? 

19.  What  part  of  a  pound  is   3*.  Sd.  ?     4.-.  Qd.1 
2s.  Id.  ?     5s.  9d  ?     6s.  Sd.  ?     4s.  9d.  ? 

20.  What  part  of  a  shilling  is  &±d.  ?  6f  c/.  ?  9Jef.  ? 

21.  What  part  of  a  pound  is  2oz.  ?     4oz.  ?     602;.  ? 


22.  What  part  of  a  quarter  is  Sib.  ?     What   part 
of  a  quarter  is  12/6.  ?     14/6.  ?     18/6.  ?     20/6.  ? 

23.  What  part  of  \cwt.  is  3  quarters  ?     2  quarters  1 
1  quarter? 

24.  What  part  of  lew*,  is  15/6.?     27/6.?     95/6.? 
75/6.  ?     68/6.  ? 

25.  What  part  of  a  ton  is  Scwtf.  ?  12cz^.  ?  14cw^  ? 
\Qcwt.  ?     19ctttf.  ? 

26.  What  part  of  a  ton  is  65/6.  ?     What  part  of  a 
ton  is  95/6.?     350/6.?     1000/6.? 

27.  What  part  of  a  foot  is  5  inches  ?     7  inches  ? 
9  inches  ?     4  inches  ?     10  inches?     11  inches? 

28.  What  part  of  a  yard  is  1  foot?     What  part  is 
,l.  foot  of  2  yards  ?     What  part  of  5  yards  is  1  foot  '? 


124  INTELLECTUAL    ARITHMETIC.  [SEC.  VII. 

29.  What  part  of  a  yard  is  1  foot  ?     What  part  of 
a  yard  is  2  feet? 

*30.  What  part  of  a  furlong  is  1  rod  ?     3  rods  ?     5 
rods  ?     8  rods  ?     10  rods  ?     12  rods  ?     15  rods  ? 

31.  What  part  of  a  mile  is  7  furlongs'?     5  fur- 
'ongs?     4  furlongs]     3    furlongs?     2  furlongs?     (> 
furlongs  ? 

32.  What   part  of  a  mile  is  20  rods?     30  rods  ? 
40  rods?     50  rods?     160  rods? 

33.  What  part  of  a  square  foot  is  12  square  inches  ? 
48  square  inches  ?     100  square  inches  ? 

34.  What  part  of  a  square  yard  is  2  square  feet? 
7  square  feet?     8  square  feet?     5  square  feet? 

35.  What  part  of  a  square  yard  is  3  square  feet? 
0  square  feet  ?      1  square  foot  ?     4  square  feet? 

30.  What   part   of  a  .rood  is  4   square  rods?     8 
square  rods  ?     9  square  rods?     7  square  rods? 

37.  What  part  of  an  acre  is  4  square  rods?      10 
square  rods?     40   square  rods?      100  square  rods? 
What  part  of  an  acre  is  3  roods? 

38.  What  part  of  a  quart  is   1   pint?     2   pints  is 
what  part  of  8  quarts  ?     3  pints  is  what  part   of  5 
quarts  ? 

39.  In  wine  measure,  what  part  of  a  quart  is  i 
pint  ?     What  part  of  a  quart  is  2  gills?     3  gilTs  'i     5 
gills? 

40.  What  part  of  a  gallon  is  3  quarts?     3  pints? 
3  gills  ?     2  quarts  ?     5  pints  ?     5  gills  ? 

41.  What  part  of  a  hogshead. is  1  gallon?     2  gal- 
Jons?     8  gallons  ?     9  gallons  ? 

42.  One  pint  in  dry  measure  is  what   part  of  a 
quart  ?     What   part  of  a    peck  ?     What  part  of  a 
bushei  } 

43.  Three  pecks  is  what  part  of  a  bushel?     What 
part  of  a  chaldron? 

44.  Five    minutes    is    what     part    of    an     hour  ? 


LE8.    V.]  INTELLECTUAL    ARITHMETIC.  125 

8  seconds  is  what  part  of  a  minute  ?     What  part  of 
an  hour  1 

45.  Three   hours  is  what  part  of  a  day  ?      What 
part  of  a  week  1 

46.  Four  days  is  what  part  of  a  week?      What 
part  cf  a  month  ?     What  part  of  a  year  ? 

47.  Five  years  is  what  part  of  a  century  1     What 
part  of  a  century  is  30  years?    40  years?    6  years? 

48.  What  part  of  3  days  is  5   hours  ?     6  hours  ? 
7  hours  ?     9  hours  ?     10  hours  ? 

49.  What  part  of  7  months  is   9  weeks?     What 
part  is  8  weeks  ?     6  weeks  ? 

50.  Four  minutes  is  what  part  of  a  day  ?     Of  an 
hour  ?     Of  a  week  ?     Of  a  month  ? 

QUESTIONS. 

1.  What  will  3 1  yards  of  cloth  cost  at  4  dollars  a 
yard? 

2.  If  4  bushels  of  wheat  be  divided   equally  among 
5  men,  how  much  will  each  receive? 

3.  What  will  5|-  bushels  of  wheat  cost  at  2^  dimes 
a  peck  ? 

4.  What  will  2f  yards  of  cloth  cost  at  2£  dimes  a 
nail? 

5.  At  2|  dimes  a  yard,  what  will  be  the  cost  of 
13-f-  yards  of  muslin  ? 

6.  What  will  12|  barrels  of  wine  cost  at  7  dollars 
a  barrel  ? 

7.  A  piece  of  cloth  containing  16|  yards  is  equally 
divided  between  3  persons  :  how  much  has  each  ODC  I 

8.  If  the  twelve  months  were  of  equal  length,  how 
many  days  would  each  contain  ? 

9.  If  a  man  travel  2^  miles  in  |-  of  an  hour,  how 
far  will  he  travel  in  5  hours? 

10.  If  wheat  is  one  dollar  a  bushel,  how  much  will 
I  quart  cost  ? 

11* 


126  INTELLECTUAL    ARITHMETIC.  [SEC.   VII. 

11.  If  cloth  is  8  dollars  a  yard,  what  will   1   nail 
cost? 

12.  If  wine  is  4  dollars  a  gallon,  what  will  3  pints 
cost?     1  quart?     2  gills? 

13.  What  will  9  yards  of  cloth  cost  at  three  cents 
a  nail  ?     At  5  cents'?     At  7  cents  ? 

14.  If  1  quarter  of  a  yard  of  cloth  costs  3-}  dollars, 
what  will  4  yards  cost  ? 

15.  If  a  man  spends  -|  of  a  dollar  in  ^  of  a   day, 
how  much  will  he  spend  in  2  weeks  ? 

16.  If  f-  of  a  hogshead  of  wine  cost  54  dollars,  what 
does  the  wine  cost  a  gallon  ? 

17.  If  a  man  earns  13  dollars  a  week,  how  much 
does  he  earn  in  each  of  the  6  working  days  ? 

18.  If  a  man  earns   1|-  dollars   a  day,  how  much 
does  he  earn  in  a  month  of  26  working  days  ? 

19.  If  hay  is  15  dollars  a  ton,  how  much  is  that  pei 
\cwt.  ?     For  1  quarter? 

20.  If  10  pounds  of  hay  cost  5^-  mills,  how  much 
will  a  ton  cost  ? 


LESSON  VI. 
Per  Cent  and  Per  Centage. 

1.  What  is  1  per  cent  of  1  dollar?     What  is  2  per 
cent  of  1  dollar  ?     3  per  cent  ?     4  per  cent  ?     5  per 
cent  ?* 

2.  What  is  4  per  cent  of  50? 

3.  What  is  6  per  cent  of  200  dollars  1 

4.  What  is  4  per  cent  of  150  dollars?     Of  200 
dollars  ?  * 

*  SUGGESTION. — Per  cent  means  by  the  hundred.    Thus,  1  per 
eent,  2  per  cent,  3  per  cent,  4  per  cent,  tfcc.,  of  any  number  o* 


LIBS.    VI.]  INTELLECTUAL    ARITHMETIC.  127 

5.  What  is  9  per  cent  of  300   dollars?     Of  400 
dollars  1 

6.  What  part  of  l.is  3  per  cent  of  30  ?     Of  4  per 
cent  of  25 1 

.  7.  What  is  6  per  cent  of  60  dollars?  Of  50 
dollars? 

8.  What  is  8  per    cent  of  70  dollars?      Of  90 
dollars? 

9.  What  is  4  per  cent    of  250  dollars?     Of  45 
dollars? 

10.  What  IS  9  per  cent  of  40?     Of  50?     Of  60? 

11.  What  is  3|  per  cent  of  100  dollars  ?     Of  40 
dollars? 

12.  What  is  2|-  per  cent  of  200  dollars?     Of  60 
dollars? 

13.  What  is  4  per  cent  of  600  dollars?     Of  30 
dollars  ? 

14.  What  is  3  per  cent  of  25  dollars?      Of  30 
dollars? 

15.  A  person  has  250  dollars,  and  takes  out  2|-  per 
cent:  how  much  will  he  have  left? 

16.  What  is  8  per  cent  of  15  dollars?     Of  25) 
Of30?     Of45? 

thing,  means  that  the  number  or  thing  is  divided  into  100  equal 
parts,  and  that  1,  2,  3,  4,  cfec.,  of  these  parts  are  taken.  The 
number  of  parts  taken,  determines  the  rate  per  cent.  Thus,  the 
rates  above,  are  1,  2,  3,  4,  <fcc.,  per  ceut. 

The^part  of  the  number  taken,  is  called  the  per  centage.  Thus, 
when  "the  thing  is  1  dollar,  and  the  rate  1- per  cent,  1  cent  is 
the  per  centage ;  if  the  rate  is  2  per  cent,  the  per  centage  is  2 
eents,  etc.  Observe, 

If  ant/  number  be  divided  by  100,  the  quotient  will  be  1  per 
osnt-  of  that  number.  Hence,  to  find  the  per  ceutage  of  any 
number,  for  any  rate  per  cent.  * 

Multiply  the  given  number  by  the  rate  per  cent  a?id  cut  off  two 
fynr^ft  front  the  right  hand  of  the  product,  which  is  equivalent 
to  dividing  by  100, 


-p\ 

128  INTELLECTUAL    ARITHMETIC.  [SEC.  VIL, 

17.  What  is  10  per  cent  of  20  dollars'?  Of  251 
Of  35?  Of 40? 

13.  What  is  the  per  centage^f  75  dollars,  at  the 
rate  of  6  per  cent  ? 

19.  What  is  the  per  centage  of  80  dollars,  at  the 
rate  of  5  per  cent  ? 

20.  What  is   the  8  per  cent  of  a  piece  of  cloth, 
measuring  50  yards  1 

21.  What  is  3^    per   cent  of  a  piece  of  muslin, 
measuring  75  yards'? 

22.  What  is  20  per  centage  of  a  box  of  shoes,  con 
taining  250  pairs  ? 

23.  What  part  of  a  number  is  5  per  cent  of  that 
number  ?* 

24.  Forty  per  cent  is  what  part  of  any  number? 

25.  What  is  25  per  cent  of  60  ?     Of  50  ?     Of  40  ? 

*  ANALYSIS — Five  per  cent  of  any  number,  is  5  hundredths 
of  that  number :  hence  5  per  cent  of  a  number  being  |  ^  of 
that  number  =  2*o  °f  that  number.  The  following  table  shows 
the  per  centage  in  terms  of  the  number : 

5  per  cent  equals  Tf  •§  =  ^  of  the  number 

6  per  cent  equals  T£¥  =  ^  of  the  number 
10    per  cent  equals  jW  =  rV  of  tue  number. 
12  £  per  cent  equals  ^/o   —  s  of  th^  number. 
15    per  cent  equals  TJ/F  —  -^  of  the  number. 
20    per  cent  equals  T2^  =  £  of  the  number. 
25    per  ceut  equals  ^5-  —  A  Of  the  number. 
30    per  cent  equals  /_«_  —  T\  of  the  number. 
33£  per  cent  equals  ±% §  t=  i  of  the  number. 
37^  per  cent  equals  ^W  =  |  of  the  number. 
40    per  cent  fquals  j4^-  •=.  |  of  the  number, 
50    per  cent  equals  ^¥  =  I  of  the  number 
75    per  cent  equals    f7—  =r  |  Of  the  number. 

100    per  cent  equals    ]  £§  =  1  the  number. 


LES.  VI.]  INTELLECTUAL    ARITHMETIC.  129 

26.  What  is  10  per  cent  of  60  ?     Of  40  ?     Of  15  ? 

27.  What  is  5  per  cent  of  40  ?     Of  80  ?     Of  100  ? 

28.  What  is   15  per  cent  of  40  dollars?     Of  80 
dollars? 

29.  What  per  cent  of  any  number  is  "|  of  it?   whai 
per  cent  is  ^  of  it  1* 

30.  Five  is  what  per  cent  of  20  ?f 

31.  Six  is  what  per  cent  of  18?    Of  24?     Of30, 

32.  Ten  is  what  per  cent  of  50  ?     Of  30  ?     Of  40  ? 
Of  60  ? 

33.  Three  is  what  per  cent  of  12?     Of  15?     Of 
24?     Of  36? 

34.  Forty  is  what  per  cent  of  80  ?     Of  20  ?     Of 
10? 

35.  Fifty  is  what  per  cent  of  200  ?     Of  60  ?     Ol 
100? 

36.  Seven  is  what  per  cent  of  49  ?     Of  21  ?     Of 
56? 

37.  Eight  is  what  per  cent  of  56  ?     Of  64  ?     Of 
84? 

*  ANALYSIS. — Since  %  of  a  number  equals  y2^,  it  follows 
that  5  of  a  number  is  equal  to  20  per  cent ;  and  we  may  find 
the  20  per  cent  by  annexing  two  (fa  to  1,  and  then  dividing  by  5. 
Hence,  we  see,  that  having  written  the  per  centage  in  the  form 
of  a  fraction,  if  we  annex  two  cyphers  to  the  numerator  and  then 
divide,  the  quotient  will  express  the  rate  per  cent. 

Therefore,  the  rate  per  cent,  wrhen  the  percentage  is  ^-,  is  100 
divided  by  6,  which  gives  16§  per  cent. 

I  ANALYSIS. — Five  is  what  part  of  20  ?  (see  lesson  VI,  page 
89.)  5  is  -/Q  of  20  —  j  of  20 ;  but  ^  is  25  per  cent :  hence,  5  is 
25  per  cent  of  20. 

Hence,  to  find  the  rate  per  cent  when  the  per  centaga  and 
DUiaber  are  known  :  Divide  the  f»€r  centage  by  the  number. 


130  INTELLECTUAL    ARITHMETIC.  fSEC.  Tr- 

QUESTIONS. 

1.  A  grocer  purchased  a  bag  of  coffee  at  ten  cents 
a  pound  :  at  what  price  must  he  sell  it  a  pound,  in 
order  to  make  10  per  cent?     What  must  he  sell  It 
for,  to  make  25  per  cent?  50  per  cent? 

2.  If  a  piece  of  broadcloth,  containing  30  yaids 
uost  5  dollars  a  yard,  what  must  it  be  sold  for  to  gaifi 
20  per  cent  ?     What  will  be  the  profit? 

3.  A  grocer  bought  10  barrels  of  flour,  at  8  dollars 
a  barrel  :  what  must  they  be  sold  for,  to  gain  25  per 
cent? 

4.  If  sugar  is  bought  at  6  cents  a  pound,  what  per 
cent  will  be  gained  if  it  be  sold  at  7? 

5.  If  a  barrel  of  flour  cost  8  dollars,  what  must  it 
be  sold  for,  to  gain  5  per  cent  ? 

6.  A  merchant  finds  that  a  lot  of  goods,  which  cost 
60  dollars,  is  damaged,  and  he  sells  them  at  a  loss  of 
15  per  cent:  what  does  he  get  for  them  ? 

7.  The  price  of  a  book  is  80  cents ;   but  being  sold 
to  a  friend,  a  discount  is  made  of  20  per  cent:  what 
is  paid  for  it? 

8.  A  piece  of  cloth,  which  cost  $45,  is  somewhat 
damaged,  and  is  sold  at  a  discount  of  33£  per  cent : 
what  is  paid  for  it? 

9.  A  merchant  buys  a  chest  of  tea,  for  which  he 
pays  85  dollars;  but  finds  it  injured,  and  sells  it  at 
20  per  cent  loss  :   how  much  does  he  get  for  it? 

10.  If  a  grocery  merchant  buys  sugar  at  6  cqnts 
a  pound,  and  sells  for  8,  what  per  cent  does  he  make  ? 

11.  A  merchant  buys  a  barrel  of  sugar  for  $60 
and   sells  it  for  $80 :  what  was  the  rate  per  cent  and 
what  the  percentage  ? 

12.  A  grocer  buys  sugar  at  5  cents  a  pound  :  what 
must  he  sell  it  for  to  make  60  per  cent  ? 

13.  A  grocer  buys  sugar  at  8  cents  a  pound  :  what 
must  he  sell  it  for  to  make  25  per  cent '? 


LES    *?!!.]  INTELLECTUAL    ARITHMETIC.  131 

14.  A  grocer  buys  sugar  at  6  cents  a  pound  and 
sells  it  at"9  :  how  much  does  he  make  per  cent? 

15.  A  grocer  buys  a  bag  of  coffee  at  12  cents  a 
pound  :   what  must  he  sell  it  for  a  pound  in  order  to 
net  1G|  per  cent] 

16.  A.  bag  of  coffee  is  bought  at  10  cents  a  pound, 
nd  being  injured,  is  sold  at  8  cents  a  pound :   what 

was  the  loss  per  cent? 

17.  If  flour  cost  9  dollars  a  barrel,  what  must  it 
be  sold  for  to  give  10  per  cent,  profit  1   12^  per  cent 
profit  1     18  per  cent? 

18.  If  molasses  costs  30  cents  a  gallon,  what  must 
it  be  sold  for  to  yield  a  profit  of  20  per  cent  ? 

per  cent? 

19.  What  is  25  per  cent  of  6  ?     Of  9  ?     Of  10  ? 

20.  Nine  is  what  per  cent  of  36  ?     Of  54  ? 


LESSON    VII. 
Of  Interest. 

1.  Interest  is  an  allowance  made   for  the  use  of 
money,  and  is  generally  reckoned   at  so   much  per 
cent  for  each   year  on  the  sum   loaned,  which  sum 
is  called  the  principal.     The  allowance,  or  per  cent- 
age,  is  called  the  interest,  and  the  principal  and  in 
terest,  together,  are  called  the  amount. 

2.  What  is  the  interest  of  $100  for   1   year,  at   1 
per  cent?     At  2  per  cent?     At  3  per  cent?     At  4 
per  cent? 

3.  What  is  the  interest  of  $150  for  1  year,  at  1  per 
cent  ?     At  2  per  cent  ?     At  3  per  cent  ? 

4.  What  is  the  interest  of  $200  for  1  year  at  2  per 
cent?     At  5  per  cent  ? 

5.  What  is  the  interest  of  $160  for  1  year,  at  5  per 
eent?     At  8  per  cent? 


132  INTELLECTUAL    ARITHMETIC.  [SEC.  VII. 

6.  What  is  the  interest  of  $200  for  2  years  at  5 
per  cent  1* 

7.  What  is  the  interest  of  §200  for  1  year  at  6  per 
cent  ? 

8.  What  is  the  interest  of  $150  for  1  year  at  8  per 
cent  ? 

9.  What  is  the  interest  of  $300  for  2  years  at  4 
per  cent  ? 

10.  What  is  the  interest  of  $500  for  2  years  at  C 
per  cent? 

11.  What  is  the  interest  of  $90  for  3  years  at  2 
per  cent  ? 

12.  What  will  be  the  amount  if  $120  be  put  at  in 
terest  for  2  years  at  6  per  cent  ? 

13.  What  will  be  the  amount  if  $60  be  put  at  in 
terest  for  3  years  at  3  per  cent  ? 

14.  If  $80  be  put  at  interest  for  2  years  at  4  per 
cent,  what  will  be  the  amount  ] 

15.  What  will  be  the  amount  of  $70  for  2  years 
at  5  per  cent  ? 

1C.  What  is  the  interest  of  $320  for  4  years  at  3 
per  cent  1 

17.  What  is  the  interest  of  $260  for  3  years  at  6 
per  cent1? 

18.  What  is  the   interest   of  125   dollars    for   2 
years  at  7  per  cent  ] 

19.  What  will  be  the  amount,  if  $300  be  put  at 
interest  for  2  years,  at  the  rate  of  4  per  cent  ? 

*  ANALYSIS. — The  interest  of  1  dollar  for  1  year  at  1  per 
cent  is  1  cent ;  and  for  any  number  of  dollars,  as  many  ceats  as 
there  are  dollars  in  the  principal.  Hence  : 

Find  the  interest  -of  the  principal  for  1  year,  at  one  per  cent 
and  then  multiply  by  the  time,  in  years,  and  by  the  rate  per  cent- 
tht  product  will  he  the  interest. 

Thus,  the  interest  of  $200  for  1  year,  at  1  per  cent,  is  $2  : 
Jien,  $2X2X5  =  820,  the  interest  for  2  years,  at  5  per  cent. 

gg|f"  The  tenths  of  a  dollar  may  be  read  dimes,  and  the  hun 
dredth*,  ueijts 


LE8.  VII.]  INTELLECTUAL    ARITHMETIC.  132 

20.  What  will  bo  the  amount,  if  $250  be  put  at 
interest  at  6  per  cent  for  3  years? 

21.  What  will  be  the  amount,  if  $500  be  put  at 
interest  for  2  years  at  4  per  cent? 

22.  What  will  be  the  amount,  if  $55  be  put  at 
interest  for  5  years  at  7  per  cent  ? 

23.  What  will  be  the  interest  of  $85  for  2  years 
at  7  per  cent  ? 

24.  What  will  be  the  interest  of  $75  at  6  per  cent 
for  5  years  ? 

25.  What  will  be  the  interest  of  $275  at  6  per 
cent  for  4  years? 

26.  What  will  be  the  amount  of  $175,  after  draw 
ing  interest  for  3  years  at  5  per  cent  ? 

27.  What  will  be  the  interest  of  $160  for  2  years 
at  8  per  cent  ? 

28.  What  will  be  the  interest  of  $375  for  2  years 
at  the  rate  of  5  per  cent  ? 

29.  What  will  be  the  amount  of  $350,  drawing  in 
terest  for  2  years  at  the  rate  of  4  per  cent  ? 

30.  What  will  be  the  amount  of  $95  for  2  years 
at  the  rate  of  5  per  cent  1 

31.  What  will  be  the  amount  of  $86  for  3  years 
at  the  rate  of  3  per  cent  ? 

32.  What  is  the  interest  of  $150  for  3  years  at  the 
rate  of  6  per  cent  ? 

33.  What  is  the  amount  of  $360  for  5  years  at  2 
per  cent? 

34.  What  is  the  amount  of  $240  for  3  years  at  3 
per  cent  ? 

35.  What  will  be  the  amount  of  $120  for  4  years 
at  the  rate  of  5  per  cent  ? 

36.  What  will  be  the  amount  of  $240  for  3  year? 
at  5  per  cent  3 


12 


134  INTELLECTUAL    ARITHMETIC.  [SEC.   V1L 

LESSON     VI. 
Interest  for  parts  of  a  Year. 

1.  What  is  the  interest  of  200  dollars  for  3  months, 
fit  5  per  cent  ?*  For  5  months  ? 

2.  What  is  the  interest  of  60  dollars  for  4  months, 
ftt  5  per  cent? 

3.  What  is  the  interest  of  $40  for  G  months  at  6 
per  cent1?  * 

4.  What  is  the  interest  of  $20  for  9  months,  at  3 
per  cent? 

5.  What  is  the  interest  of  $15  for  10  months,  at 
6  per  cent  ? 

G.  What  is  the  interest  of  $G  for  8  months,  at  7 
per  cent  1 

7.  What  is  the  interest  of  $12  for  11  months,  at  8 
per  cent  ? 

8.  What  is  the  interest  of  $25  for  10  months,  at  9 
per  cent  ? 

9.  What  is  the  interest  of  $60  for  5  months,  at  0 
per  cent  1 

10.  What  is  the  interest  of  $84  for  11  months  at 
10  per  cent? 

11.  What  is  the  interest  of  $96  for  7  months,- at  9 
per  cent  ? 

12.  What  is  the  interest  of  $72  for  5  months,  afc  7 
per  cent ? 

13.  What  is  the  interest  of  $144  for  11  months,  at 
9  per  cent? 


*  ANALYSIS. — Find  the  interest  for  1  year,  at  1  per  cent, 
which  is  $2 ;  then,  as  3  months  being  \  of  a  year,  the  interest 
for  three  months  will  be  3-  of  2,  or  -^  a  dollar ;  then  multiply 
by  5,  the  rate  per  cent,  and  we  obtain  2J  dollars,  or  2  dollars 
and  fifty  cents,  for  the  interest 

For  5  months,  we  have  y5^  of  2  dollars,  which  is  |-  of  one 
dollar,  which  being  multiplied  by  5,  gives  2-$  ,  or  4|-  dollars. 


LES.  VII. J  INTELLECTUAL    ARITHMETIC.  185 

14.  What  is  the  interest  of  $60  for  6  days,*  at  8 
per  cent? 

15.  What  is  the  interest  of  $84  for  9  days,  at  7 
per  cent? 

16.  What  is  the  interest  of  $24  for  16  days,  at  8 
per  cent  ?f 

17.  What  is  the  interest  of  $48  for  17  days,  at  6 
per  cent? 

18.  What  is  the  interest  of  $50  for  6  days,  at  8 
per  cent  ? 

19.  What  is  the  interest  of  $96  for  11  days,  at  7 
per  cent  ? 

20.  What. is  the  interest  of  $40  for  15  days,  at  9 
per  cent  ? 

21.  What  is  the  interest  of  $144  for  8  days,  at  5 
per  cent  ? 

22.  What  is  the  interest  of  $60  for  18  days,  at  10 
per  cent? 

23.  What  is  the  interest  of  $72  for  10  days,  at  9 
per  cent  ? 

24.  What  is  the  interest  of  $132  for  5  days,  at  6 
per  cent  ? 

25.  What  is  the  interest  of  $42  for  20  days,  at  9 
per  cent  ? 

26.  What  is  the  interest  of  $12  for  19  days,  at  10 
per  cent  ? 

27.  What  is  the  interest  of  $36  for  21  days,  at  10 
per  cent? 

28.  What  is  the  interest  of  $15  for  25  days,  at  8 
per  cent? 

*  ANALYSIS. — The  interest  of  $60  for  1  year,  at  1  per  eent  is 
30  cent1! ;  and  for  1  month,  is  5  cents,  and  for  1  day  ia'-fo  r* 
|  of  1  cent ;  and  at  8  per  cent,  itis-g-X8~|-—  l£  cents 

f  OBSERVATION. — Observe  that  16  days  is  £  of  a  month,  and  1 
Jay  over ;  then  as  the  interest  at  1  per  cent  is  2  cents  a  month. 
16  days  gives  1  cent,  and  ^  of  2  cents,  or  y1^  of  a  cent:  hence 
IT*S  of  a  cent  multiplied  by  8,  gives  8j8g  of  a  cent 


136  INTELLECTUAL    ARITHMETIC.  [SEC.  VIJ 

29.  What  is  the  interest  of  $20   for  2  years   £ 
months  and  6  days,  at  7  per  cent  ? 

30.  What   is  "the  interest  of  $30  for  3  years  9 
months  and  10  days,  at  6  per  cent? 

31.  What   is   the  interest   of  $24  for  4  years  0 
months  and  20  days,  at  8  per  cent  ? 

32.  What  is  the  interest  of  $36  for  1  year  and    5 
days,  at  8  per  cent? 

33.  What  is  the  interest  of  60  dollars  for  2  years 
3  months  and  25  days,  at  6  per  cent? 

31.  What  is  the  interest  of  $200  for  2  years  10 
months  and  10  days,  at  5  per  cent? 

35.  W^hat  is  the  interest  of  60  dollars  for  2  years 

2  months  and  3  days,  at  8  per  cent? 

36.  What  is  the  interest  of  48  dollars  for  3  years 

3  months  and  10  days,  at  2  per  cent? 

37.  What    is   the  interest  of  $72  for  3  years  6 
months  and  5  days,  at  4  per  cent  ? 

38.  What   is   the  interest  of  $84  for  2  years  4 
months  and  3  days,  at  6  per  cent? 


LESSON    VIII. 

Analysis  of  Questions. 

1.  If  1   yard    of  cloth  cost  $2,  how  much  will  4 
yards  cost  at  the  same  rate? 

2.  If  5  yards  of  cloth  cost  $10,  what  will  8  yards 
cost  at  the  same  rate  ? 

3.  If  4  yards  of  cloth  cost  $9,  what  will  16  yards 
cost  at  the  same  rate  ?* 

*  ANALYSIS. — This,  and  similar  examples,  may  be  done  by  tli« 
analysis  on  page  53  ;  or  they  may  be  done  thus : 

The  ratio  of  4  yards  of  cloth  to  16  yards  is  4 ;  that  is,  16 
yards  of  cloth  is  4  times  as  much  as  4  yards,  and  therefore,  will 
cost  4  times  as  much  ? 

If  4  yards  of  cloth  cost  $9,  16  yards  will  cost  4  times  9  del 
lars,  which  are  36  dollars. 


&9ES.  VIII.]  INTELLECTUAL    ARITHMETIC.  137 

4.  If  6  men  consume  a  barrel  of  flour  in  2  months, 
how  much  will  they  consume  in  a  year? 

5.  If  a  man  travels  60  miles  in  5  days,  how  far 
will  he  travel  in  30  days  1 

6.  If  4  men  consume  1  barrel  of  flour  in  20  days, 
how  much  would  32  men  consume  in  the  same  time] 

7.  If  3  barrels  of  flour  cost  $14,  how  much  will  0 
barrels  cost  1 

8.  If  4/6.  of  sugac  cost  64  cents,  what  will   13/6. 
cost? 

9.  If  |  of  a  piece  of  cloth  costs  $8J,  what    will   f 
pieces   cost  1 

10.  If  £  of  a  barrel  of  cider  cost  T\  of  a  dollar, 
what  will  |  of  a  barrel  cost? 

11.  If  9  bushels  of  oats  will  feed  4  horses  5  days, 
how  long  will  36  bushels  feed  them  ? 

12.  If  3  paces  of  the  common  step  be  equal   to  2 
yards,  to  how  many  yards  will  18  paces  be  equal? 

13.  If  5  yards  of  cotton  cloth  are  equal  in  value  to 
2  yards  of  linen,  how  many  yards  of  linen  will  20 
yards  of  cotton  buy  ? 

14.  If  8/6.  of  coffee  is  of  the  same  value   as  3/6. 
of  tea,  how  many  pounds  of  tea   will  24/6.  of  coffee 
buy? 

15.  If  8  oranges  are  worth  24  cents,  how  much  are 
2  oranges  worth? 

16.  If  4  apples  are  worth    1^-  oranges,  and   one 
orange  is  worth  two  lemons,  how  many  lemons  will 
12  apples  buy  ? 

17.  If  5   baskets  of  peaches  are  worth  $8f ,  how 
much  will  8  baskets  be  worth  ? 

18.  If  a  man  travel  7  miles  in  two  hours,  how  fai 
will  he  travel  in  14  hours  ? 

19.  Two  men  start  from  the  same  point  and  travel 
in  opposite  directions ;  one  at  the  rate  of  3-J  miles  an 
hour,   and   the   other  at    the    rate  of  4*    miles  aD 

12* 


138  INTELLECTUAL    ARITHMETIC.  [SEC.  VII. 

»jin"^  L 

hour  ;  how  far  apart  will  they  be  at  the  end  of  4 
hours  ? 

20.  If,  in  the  last  question,  the  men  were  to  travel 
in  the  same  direction,  how  far  apart  would  they  be 
at  the  expiration  of  4  hours  ? 

21.  If  6/6.   of  butter  cost  14  shillings,  how  mueft 
will  15/6.  cost? 

22.  If  4  turkeys  cost  $2f ,  what  will   16  turkeys 
cost? 

23.  If  9 'yards  of  broadcloth  cost  $27^-,  how  much 
will  27  yards  cost? 

24.  If  5/6.  of  loaf  sugar  cost  $-f,  how  much  will 
twenty-five  pounds  cost? 

25.  If  2  tons  of  hay  cost  19J  dollars,  what  will  8 
tons  cost? 

26.  If  3  pairs  of  boots  cost  $21^  dollars,  how  much 
will  18  pairs  cost  ? 

27.  If  3  horses  eat   6-|   bushels  of  oats  in  2  days, 
how  much  will  12  horses  eat  in  the  same  time? 

28.  If  a  family  consume  1|  barrels  of  flour  in  2 
months,  how  much  will  they  consume  in  4  months? 

29.  If  7  bushels  of  wheat  cost  $6f,  how  much  will 
14  bushels  cost? 

30.  If  a  family   consume  2-|  bushels  of  grain  in 
3|-   weeks,    how    much    will    they    consume    in    6-J 
weeks  ? 

31.  If  3£  yards  of  cloth  cost  $6f,  what  will   14 
yards  cost  ? 

32.  If  3  pairs  of  shoes  cost  $5f ,  what  will  9  pairs 
cost? 

33.  If  a  man  travels  9  miles  in  2J  hours,  how  far 
will  he  travel  in  7£  hours  ? 

34.  If  9i  pounds  of  tea  cost  12|  dollars,  how  much 
will  28  pounds  cost? 

35.  If  8  yards  of  broadcloth  cost  17  dollars,  what 
will  16  yards  cost? 


UKS.  IX.]  INTELLECTUAL   ARITHMETIC.  139 

LESSON   IX 

Analysis  of  Questions  Continued. 

1.  If  a  man  can  build  a  wall  in  1  day,  how  long 
will  it  take  two  men  to  build  it  ? 

2.  If  2  men  can  build  a  wall  in  4  days,  how  long 
will  it  take  4  men  to  build  it  ? 

3.  If  4  horses,  in  two  days,  eat  5  bushels  of  oats, 
how  much  will  6  horses  eat  in  4  days'?* 

4.  If  a  barrel  of  flour  last  15  men  20  days,  how 
long  will  it  last  25  men  ?f 

5.  If  6  men  consume  24$.  of  beef  in  5  days,  how 
much  will  9  men  consume  in  ten  days  ? 

6.  If  6  horses  eat  2-j  tons  of  hay  in  2  weeks,  how 
much  will  16  horses  eat  in  1-|-  weeks  1 

7.  If  8  men.  can  build  a  wall  6  days,  in  how  many 
days  can  12  men  build  it? 

8.  If  a  certain  amount  of  provisions  will  last  2 
families  of  5  persons  each  3  weeks,  how  long  will  the 
same  provisions  last  5  families  of  6  persons  each  1 

9.  If  2  men,  in  5  days,  can  build  160  feet  of  wall, 
how  long  will  it  take  4  men  to  build  192  feet  of 
wall? 

10.  If  5  men  can  do  a  certain  work  in  6  days,  how 
long  will  it  take  3  men  to  do  5  times  that  work  ? 

*  ANALYSIS — Four  horses  will  eat  as  much  in  2  days  as  8 
horses  in  1  day ;  and  6  horses  will  eat  as  much  in  4  days  as  24 
horses  in  1  day.  ]STow,  24  horses  is  3  times  8  horses:  if  8 
horses  eat  6  bushels  of  oats  in  1  day,  24  horses  will  eat  3  times 
as  much,  which  are  15  bushels:  therefore,  6  horses  in  4  days 
{equivalent  to  24  horses  for  1  day)  will  eat  15  bushels. 

f  Fifteen  men  will  eat  as  much  in  1  day  as  1  man  will  eat  ia 
15  days  and  15  men  will  eat  as  much  in  20  days  as  1  man 
in  300  days.  Now,  the  same  provisions  will  last  25  men  only 
one-twenty-fifth  as  long-  as  they  will  last  1  man;  that  is,  so  many 
days  as  25  is  contained  times  in  300,  which  are  12. 


140  INTELLECTUAL    ARITHMETIC.  [SEC.  VH, 

11.  If  a  man  travels  48  miles  in  2  days,  travelling 
6  hours  a  day,  how  far  will  he  travel  in  3  days,  travel 
ling  at  the  same  rate,  5  hours  a  day  ? 

12.  If  12  dollars'  worth  of  provisions  will  supply  9 
men  4  clays,  how  much  will  it  cost  to  supply  21  men 
for  5  days? 

13.  If  2  men  consume  2$.  4oz.  of  flour  in  1  day 
h(,w  much  will  8  men  consume  in .4  days? 

14.  How  many  sheep,  at  3  dollars  a  head,  mustb© 
given  for  5  cows  at  $18  apiece? 

15.  A  man,  failing  in  trade,  pays  his  creditors  3 
shillings  on  the  dollar,  while  another  pays  an  equal 
gum  by  paying  two  shillings  on  the  dollar:  what  ifr 
the  ratio  of  their  debts? 

1G.  If  |  of  a  bushel  of  oats  will  feed  2  horses  half 
a  day,  how  many  will  be  required  to  feed  4  horses  4-J 
days? 

17.  If  a  barrel  of  flour  will  serve  a  family  of  G  per 
sons  3;}  weeks,  how  much  will  serve  a  family  of  9  per 
sons  4|-  weeks? 

18.  if  5  men  can  cut  30  cords  of  wood  in  3  days, 
how  much  will  4  men  cut  in  8  days  ? 

19.  If  3  men  can  mow  15  acres  in  2£  days,  how 
long  will  it  take  11  men  to  mow  44  acres? 

20.  If  a  farmer  can  plough  9  acres  in  4  days,  with 
one  team,  how  much  can  he  plough  with  two  teams 
in  8  days? 

21.  If  a  pasture  of  8  acres  will  feed  3  horses  for 
2  months,  how  many  acres  will    feed    4    horses   5 
months  ? 

22.  If  the  wages  of  3  men  for  7  days  are  21  dollars, 
what  will  be  the  wages  of  9  men  for  11  days  ? 

23.  If  a  man  travels  12-J  miles  in  5  hours,  how  far 
will  he  travel  in  4  hours,  at  the  same  rate  ? 

24.  A  man  loses,  at  play,  y  of  his  money,   after 
which  he  gives  away  ^  of  the  remainder,  and  finds 
that  he  has  8  dollars  left ;  how  much  had  he  at  first  if 


LES.    X.J  INTELLECTUAL    ARITHMETIC.  141 

25.  How  many  yards  of  cloth  at  4-J-  dollars  a  yard, 
must  be  given  fur  9  yards  at  3  dollars  a  yard  ? 

26.  If  4  tailors  can  make  8  pair  of  pantaloons  in  2 
days,  how  many  will  3  tailors  make  in  5  days] 

27.  If  5  horses  consume  24|  tons  of  hay  in  a  winter, 
how  much  will  10  horses  consume? 

28.  James  being  asked  how  many  marbles  he  had, 
replied,  that  if  f  of  the  number  be  divided  by  2,  and 
the  quotient  subtracted  from  one  half  the  number,  the 
remainder  would  be  12  :  how  many  had  he  ? 

9  29.  If  9  men  can  do  a  piece  of  work  in  4|  days, 
how  many  men  should  be  employed  to  do  the  same 
work  in  7  days'? 

30.  If  3  horses  eat  3-|  tons  of  hay  in  2  months,  how 
much  will  supply  5  horses  for  4  months  1 


LESSON    X. 

To  find  the  parts,  knowing  the  whole  and  the  proportion  of  the 
parts. 

1.  James  bought  an  orange  and  a  melon,  for  which 
he  paid  8  cents.     He  paid  three  times  as  much  for 
the  melon  as  for  the  orange:  what  did  he  pay  for 
each  ?* 

2.  Charles  bought  a  whistle  and  a  top,  for  which 
he  paid  12  cents.     He  paid  five  times  as  much  for 
the    whistle  as  for    the    top  :    what    did  he  pay  for 
each? 

3.  What  number  added  to  itself  will  give  a  sum 
equal  to  20  ] 

4.  What  number  added  to  twice  itself  will  give  a 
number  equal  to  15  ? 

*  ANALYSIS — James  paid  one  equal  part  of  the  whole  sum  fo/ 
the  orange,  anu  3  equal  parts  for  the  melon :  hence,  he  paid  4 
tqujl  j>art*  for  both.  Then,  the  whole  sum  which  he  paid.  (8 
cents),  divided  by  the  number  of  equal  parts,  (4  equal  parts), 
will  give  1  part,  which  is  2  cents,  what  he  paid  for  the  orange; 
and  2  cents  multiplied  by  3,  will  give  6  cents,  what  he  paid  for 
Hie  meicu. 

5 


142  INTELLECTUAL    ARITHMETIC.  [SEC.  VII. 

5.  What  number  added   to  five  times  itself  will 
give  a  number  equal  to  30  1 

6.  John  bought  an  apple,  a  peach,  and  an  orange, 
for  which  he  paid  6  cents.     He  paid  twice  as  much 
for  the  peach   as   for  the  apple,  and  as  much  for  the 
orange  as  for  the  apple  and  peach  together  :  what  did 

e  pay  for  each  ? 

7.  A  man  bought  a  horse,  saddle,  and  bridle,  for 
which  he  paid   90  dollars.     He  paid  twice  as  much 
for  the  saddle  as  for  the  bridle,  and  four  times  as  nnich 
for  the  horse  as  for  the  saddle  and  bridle  together  : v 
what  did  he  pay  for  each.?     How  many  parts  are 
there  ? 

8.  The  sum  of  the  ages  of  James,   Charles,  and 
John,  is  44  years.     James'  age  is  one  half  Charles 
and  one  third  John's  :  what  is  the  age  of  each  ? 

9.  A  farmer  has  in  his  garden  apple  trees,  pears, 
and  peaches  ;  in  all  72.     He  has  twice  as  many  pear 
as  apple  trees,  and   three   times  as  many  peaches  as 
pear  trees  :  how  many  has  he  of  each  ? 

10.  A  gentleman  gave  06  cents  to  a  father,  mother, 
and  boy.     He  gave  the  mother  twice  as  much  as  the 
boy.  and  the  father  twice  as  much  as  the  mother  and 
boy  together  :  how  much  did  he  give  to  each  :l 

11.  A  farmer  has  72  sheep  in   four  lots.     In  the 
second  he  has  twice  as  many  as  in  the  first ;  in  the 
third  as   many  as  in   the  second  ;  and  in  the  fourth 
twice  as  many  as  in  the  third :  how  many  has  he  in 
each  ? 

'  12.  Mary,  Jane,  and'  Nancy,  gather  144  apples 
from  the  orchard.  Jane  is  to  have  twice  as  many  as 
Mary,  and  Nancy  is  to  have  three^  times  as  many  as 
Mary  and  Jane  together  :  how  many  will  each  have! 
13.  Divide  12  into  two  such  parts  that  the  second 
shall  be  double  the  first.  Into  how  many  equal  parts 
is  12  to  be  divided? 


LEd.     X.J  INTELLECTUAL    ARITHMETIC.  143 

14.  Divide   21    into    three    such    parts    that   the 
second  shall  be  double  the  first,  and  the  third  double 
the  second. 

15.  James  asked  John  how  many  marbles  he  had  ? 
John  replied,  if  you  will  give  me  twice  as  many  as  1 
now  have,  and  William  will  give  me  5  times  as  many 
as  I  should  then  have,  I  would  have,  in  all,  36  :  how 
many  had  he  1 

16.  Mr.    Parsons    bought    4    pounds  of  coffee,   a 
pound  of  tea.  and  a  yard  of  cloth,  and  paid  in  all  $16. 
He  paid  twice  as  much  for  the  tea  as  for  the  coffee, 
and  5  times  as  much  for  the  cloth  as  for  the  coffee : 
what  did  he  pay  for  each? 

17.  Divide  36  into  four  such  parts  that  the  second 
shall  be  3  times  the  first,  the  third  5  times  the  first, 
and  the  fourth  9  times  the  first. 

18.  m  a  pasture  there  are  seven   times  as  many 
sheep  as  cows,  and  twice  as  many  lambs  as  cows  ;  m 
all,  40  :  how  many  of  each  sort  1 

19.  A  father,  mother,  and  son,  receive  108  cents 
for  a  day's  work.      The   mother    receives    twice   as 
much    as    the   son,   and  the  father   as   much   as   the 
mother  and  son  together  :  what  does  each  receive  1 

20.  In  a  school   of  three  departments,   there   are 
150  pupils.     In  the  first  department  there  are  one 
third  as  many  as  in  the  second,  and  in  the  second  one 
half  as  many  as  in  the  third  :  how  many  are  there  in 
each  ]     How  many  equal  parts  of  the  whole  school 
in  each  department  ? 

21.  Divide  82  into  four  such  parts  that  the  second 
shall  be  4  times  the  first,  the  third  3  times  the  second., 
and  the  fourth  2  times  the  third. 

22.  A  slooj)  employed  in  carrying  bricks  to  New 
York,  carries  40  thousand  at  a  load.     She  is  loaded 
twice  and  ur loaded  once,  the  first  week — the  second 
week,  she  is  unloaded  twice  and  loaded  once  ;  and  se 


144  INTELLECTUAL    ARITHMETIC.  [SEC.  VII, 

on  for  the  season  :   how  much  does  she  average  a 
week  ? 

23.  James  being  asked  what  he  had  been  about 
during  the  day,  replied,  that  he  had  been  ciphering  4 
hours  and  done  82  sums.  That  in  the  second  hour 
he  did  4  times  as  many  as  in  the  first;  in  the  third 
hour,  three  times  as  many  as  in  the  second  ;  and  in 
the  fourth,  2  times  as  many  as  in  the  third  :  how 
naany  did  he  do  in  each  hour  ] 


LESSON     XI. 
To  divide  Numbers  into  proportional  parts. 

1.  Divide  the  number  18  into  two  parts,  such  that 
tile  ratio  of  the  parts  shall  be  the  same  as  2  to  4.* 

2.  Divide   the  number  28    into    two    parts,  such 
that  their  ratio  shall  be  the  same  as  5  to  0. 

3.  Divide  the  number  34  into  two  parts,  such  that 
the  first  shall  be  eight  ninths  of  the  second. 

4.  Divide   34  into   two   such  parts  that  the   first 
shall  be  one  and  one  eighth  times  the  second. 

5.  Two  men  bought  a  piece  of  muslin  containing 
30  yards ;  one  paid  $2  and  the  other  $3  :  how  many 
yards  belonged  to  each  1 

6.  Two  men  hired  a  pasture  for  $24.     One  pastured 
5  horses  and   the  other  3 :  how   much  should  each 
pay? 

7.  Two  men  hired  a  pasture  for  $72.    One  pastured 
3  horses  for  5   weeks,  and   the  other  7  horses  for  3 
weeks:  what  proportion  should  each  pay  1 

8.  A  father   divides  84  cents  between  John  and 
diaries,  giving  5  cents  to  John  .and  7  to  Charles  each 

*  ANALYSIS. — Theiv  are  6  units  in  the  sum  of  2  and  4.  If  18 
be  divided  into  6  equal  parts,  each  part  will  be  3.  IVo  of 
tlie»e  parts  must  form  the  tirst  number,  and  4  of  them  the 
second.  Hence,  the  numbers  are  6  and  12. 


LES.    XI.]  INTELLECTUAL    ARITHMETIC.  145 

lime,  till  the  whole  was  distributed  :  how  much  did 
lie  give  to  each] 

9.  Three  persons  buy  a  piece  of  cloth  containing 
48  yards.     The  first  puts  in  5  dollars,  the  second  9, 
and  the  third  10:  what  was  each  one's  share'? 

10.  James  has  72  marbles  ;  he  gives  3  to  William 
and  5  to  John,  each  time,   until  none  are  left:    how 
many  does  he  give  to  each  1 

11.  William  has  9  cents  and  John  7,  and  they  buy 
D6  apples  :  how  many  apples  should  each  have? 

12.  Mr.  Wilson  fails  in  business,  and  pays  J  «f  hi? 
entire  debts :  how  much  should  Mr.  Squires  receive,  to 
whom  he  owes  $108? 

13.  A  grocer  weighs  out  24  pounds  of  sugar  to  2 
customers,  giving  2  pounds  to  one  as  often  as  he  gave 
I  of  a  pound  to  the  other :  how  much  did  he  give  to 
each? 

14.  A  draper  divides  a  piece  of  cloth  containing  30 
yards,  between  2  persons,  giving  2|  yards  to  the  one 
every  time  that  he  gave  3-J-  yards  to  the  other:  how 
much  did  each  receive? 

15.  A  man  distributed  78  cents  among  8  beggars, 
3    of    whom  were  men    and   5   were    women.       He 
gave  twice  as  much  to  each  woman  as  to  each  man: 
how  much  did  he  give  to  each  ? 

16.  James  and  John  start  from  the  same  place, 
travel  the  same  way,  and  take  steps  of  equal  length 
James  steps  4  times  while  John   steps  but  3:  how 
far  will  they  be  apart  when  the  djstances  travelled  by 
both  is  14  miles? 

17.  James  and  John   start  from   the  same  place, 
travel  the  same  way,  and  take  steps  of  equal  length; 
James  steps  4  times  while  John   steps   but  3  :   hov* 
far  will  each  have  travelled  when  they  are  3  miles 

apart  ? 

13 


146  INTELLECTUAL    ARITHMETIC.  [SEC.  VII. 

18.  Two  men  agree  to  do  a  piece  of  work  for  which 
they  are  to  receive  $88  ;  the  first  sends  4  hands  for  3 
days,  and  the  second  5  hands  for  2  days:  how  much 
should  each  receive  ? 

19.  A  person  met  three  beggars,  a  boy,  a  mother 
and  father,  and  distributed  84  cents   among   them. 
For    every    5    cents   he    gave   the  boy  he  gave  the 
mother  7  and  the  father  9  :  how  much  did  he  give. to 
each  ? 

20.  Three  persons  hire  a  pasture,  for  which   they 
pay  $56.     The  first  puts  in  2  horses  for  3  weeks,  the' 
second  5  horses  for  2   weeks,  and  the  third  9  horses 
for  li  weeks  :  how  much  ought  each  to  pay  1 

21.  Charles  has  5  marbles  and  John  9,  and  they 
agree  to  share  their  winnings  or  losses  in  the  same 
proportion.     After  several  games  they  find  that  they 
have  won  42  :  how  are  they  to  be  divided  ? 

22.  A  and  B  enter  into  partnership  :  A  puts  in  7 
dollars,  and  B  11  :  they  make  9  dollars  by  the  opera 
tion  :  how  should  it  be  divided  ? 

23.  A  and  B  enter  into  partnership  :  A  puts  in  0 
dollars  for  2  months,  and  B,  5  dollars  for  3  months: 
they  gain  81  dollars  :  what  is  the  share  of  each? 

24.  Two  barrels  of  flour,  costing   12   dollars,  are 
consumed  by  three  persons ;  the  first  ate  from  them 
2  months,  the  second  3    months,  and    the    third   5 
months  :  how  much  should  be  paid  by  each  ? 

25.  Three  persons    hire  a   pasture  for  sheep,  for 
which  they  pay  12  dollars.     The  second  puts  in  twice 
as  many  sheep  at  the  first,  and  the  third  three  times 
as  many  as  the  first ;  but  the  sheep  belonging  to  the 
first  man  are  in  twice  as  long  as  those  belonging  to 
the  second,  and  three  times  as  long  as  those  belonging 
to  the  third  :  how  much  should  each  pay  1 


LE3.      XII.]  INTELLECTUAL    ARITHMETIC.  147 

LESSON   XII. 
Analysis  of  Questions. 

1.  The  sum  of  two  numbers  is  10  and  their  dif 
ference  4:  what  are  the  numbers?* 

2.  The   sum  of  two   numbers  is  16  and  their  dif 
ference  8  :  what  are  the  numbers  ? 

3.  James  and  John  together  have  24  marbles,  and 
the  difference  between  their  marbles  is  one  fourth  of 
the  sum  :  how  many  had  each  ? 

4.  A  tailor  in  measuring  two  pieces  of  cloth  found 
their  difference  to  be  6  yards,  and   also    that  this  dif 
ference  was  an  eighth  part  of  the  cloth  in  both  pieces  : 
how  much  was  there  in  each  piece  ? 

5.  James   and   John  have  16   marbles,  and  Jamecs 
has  4  more  than  John :   how  many  has  each  ? 

6.  Nancy  has  6  more  pins  in  her  cushion  than  Jane, 
and  together  they  have  30  :  how  many  has  each  ? 

7.  John,  in  a  week  recited   10   lessons    more    than 
Charles,  and  together  they  recited  24:  how  many  did 
each  recite  ? 

8.  A  farmer  bought   an   equal   quantity   of  sugar 
and  coffee,  and  then  gave  a  cheese  for  20  poundt  of 
coffee,  when  it  appeared  that  he  had  in  ail  50  pounds 
of  sugar  and  coffee:   how  much  had  he  of  each  ? 

9.  A  man   being  a^ked   how  much  money  he  had, 
said,  that  he  had  only  dollars  and  dimes,  and  that  he 
had   72  pieces   in  all:    that  the    number,  of  dollars 
less  the  number  of  dimes,  was  one-twelfth  the  sum 
of  the  pieces:   how  much  money  had  he? 

*  ANALYSIS. — If  the  two  numbers  were  equal  and  their  sum 
10,  each  number  would  be  5.  Now,  if  you  take  1  from  one  of 
Hie  5's  you  have  the  number  4,  and  if  you  add  it  t.o  the  other 
you  get  the  number  6  ;  if  you  do  ilie  same  for  the  two  last  num 
bers,  you  get  the  numbers  3  and  7,  whose  difference  is  4  :  that  i?, 
the  greater  of  two  numbers  in  equal  to  half  /heir  xnin.  plus  half 
their  difference,  and  the  less  is  equal  to  half  their  sum  minus 
half  their  difference.  • 


148 


INTELLECTUAL    ARITHMETIC.  [SEC.  VII, 


10.  A  farmer  has  10  more  sheep  than  he  has  cows, 
he  loses  3  cows  and  6  sheep,  when  he  finds  that  he 
has  17  of  both  kinds  remaining:  how  many  had  heat 
first  of  each  kind  1 

11.  A   father  being  asked   his  age,  replied,  that  7 
years  ago  his  age  was  double  that  of  his  son's,  and 
now  that  the  sum  of  their  ages  was  89  :  what  was  the 
age  of  each  ? 

12.  A  man  and  son  engage  to  work  20  days,  the 
son  to  receive  3  dimes  a  day  less  than  the  father: 
at  the  end  of  the  time  they  receive  42  dollars  :   how 
much  of  this  sum  did  each  earn,  and  what  did  each 
receive  per  day  ? 

13.  John  being  asked  how  many  marbles  he  had, 
replied,  that  19  was  3  more  than  J-  of  the  number: 
how  many  had  he  1 

14.  Lucy  being  asked  her  age,  said,  that  her  sister 
Jane  was  6   years  old  when   she  was  born,  and  that 
now  the  sum  of  their  ages  was  20 :  what  was  the  age 
of  each  ? 

15.  A  former  had  as  many  sheep  as  hogs,  and  after 
losing  12  of  his  hogs,  his  sheep  and  hogs  amounted  to 
00  :  how  many  had  he  at  first  of  each  kind  ? 

16.  A  man   bought  a   vest,  for  which    he  paid   21 
dollars  less  than  he  paid  for  his  coat,  and  for  the  two 
together  he  paid  33  dollars:  what  did  he  pay   for 
each  ? 

17.  Mr.  Wilson  sold  his  cow  for  30  dollars,  which 
was  -}^  of  what  she  cost  him  :   what  did  he  give  for 
her? 

18.  A  gentleman  bought  a  coat  and  hat,  for  which 
he  paid  27  dollars,  and   the  cost  of  the  hat  was  one 
eighth  of  the  cost  of  the  coat:   what  was  the  cost  of 
each  ? 

19.  If  f  of  a  number  is  3  less  than  g,  what  is  the 
number! 


LES.   XIII.]  INTELLECTUAL    ARITHMETIC.  149 

20.  Two  men  have  60  dollars  between  them  ;  if  the 
!.me  having  the  largest  sum  gives  the  other  5  dollars, 
their  money  will  be  equal :  how  much  had  each  ? 

21.  A  man  bought  a  cow  and  a  calf,  for  which  he 
paid  36  dollars,  paying  5  times  as  much  for  the  cow 
as  for  the  calf:  what  did  he  pay  for  each  ? 

22.  A  man  bought  a  pair  of  boots  for  6f  dollars, 
which  was  |-  of  what  he  paid  for  his  coat :  what  did  his 
coat  cost  ? 

23.  A  man  paid  2|  dollars  more  for  his  pantaloons 
than  for  his  vest,  and  for  both  he  paid   141  dollars: 
what  did  he  pay  for  each  ? 

24.  A  market  woman*  bought  36   eggs ;   for  f  of 
them   she  paid   2  cents  for  three  eggs,  and   the  re 
mainder  she  bought  at  the  rate  of  4  cents  for  3  eggs  : 
for  what  must  she  sell   them   that  she  may  make  G 
cents? 


LESSON    XIII. 
Separate  and  Concurring  Causes. 

1.  If  Charles   can   do  a  piece  of  work  in  2  days, 
what  part  of  it  can  he  do  in  1  day  ? 

If  you  denote  the  work  by  1,  what  will  denote  the 
part  which  he  can  do  in  1  day  1 

2.  If  James  can  do  a  piece  of  work  in  3  days,  what 
part  of  it  can  he  do  in  1  day  ?     What  part  in  2  days  ? 

3.  If  a   family  consume   12  pounds  of  sugar  in  a 
week,  how  much  will  they  consume  in  1  day  1     How 
much  in  3  days  ?     4  days  ? 

4.  If  James  can  do  one  fifth  of  a  piece  of  work  in 
1  day,  how  long  will   it  take   him   to   do   the  entire 
work?     If  he  can  do  \  in  a  day,  how  long  will  it  take 
him  to  do  the   work  ?     How  long  if  he  can  do  £  ? 
If  he  can  do  •£,  how  long  ? 

5.  John  can  do  a  piece  of  work  in   2  days,  arid 
Charles  can  do  the  same  work  in  3  days : 

13* 


150  INTELLECTUAL    ARITHMETIC.  [SEC.   VIL. 

What  paft  of  the  work  can  each  do  in  a  single  day  1 

AY  hat  part  can  both  do  in  I  day  ;  and  iv«  what 
time  can  the  work  be  done  by  both  of  them  working 
together?* 

0.  A  can  do  a  certain  piece  of  work  in  3  days,  B 
can  do  it  in  5  days  : 

AY  hat  part  of  it  can  each  do  in  1  day  ?  What  part 
can  both  do  in  1  day,  and  in  what  time  can  the  work 
be  done  by  both  working  together? 

7.  A  cistern  is  to  be  filled  by  two  pipes.     One  can 
(ill  it  in  2  hours  and  the  other  in  5 : 

What  part  of  the  cistern  can  each  fill  in  1  hour  1 
What  part  of  it  can  they  botn  till  in  1  hour  ?  In  what 
time  can  they  fill  the  cistern  running  together  ? 

8.  A  cistern  is  to  be  filled  by  three  pipes.     The 
first  can  fill  it  in  2  hours,  the  second  in  3,  and  the 
third  in  4 : 

What  part  of  it  will  each  fill  in  1  hour?  What 
part  of  it  will  they  all  fill  in  1  hour?  In  what  time 
will  the  three  fill  it,  running  together? 

9.  John  can  do  a  piece  of  work  in  2  days,  and  John 
and  James  together,  can  do  it  in  1  j-  days :  in  what 
time  can  James  do  it  alone  ?•(• 

*  ANALYSIS. — Denote  the  work  to  be  done  by  1.  Then  -J-  will 
will  represent  the  part  which  John  can  do  in  1  day  ;  and  ij  will 
represent  the  part  which  Charles  can  do  in  1  day  ;  and  the  sum 
of  \  and  3-  will  represent  what  both  can  do  in  1  day,  viz. :  |-  of 
the  entire  work.  Then,  the  number  of  times  which  1  (the  work 
to  be  done)  contains  -f ,  viz.:  \\  times, shows  the  number  of  days 
in  which  they  can  do  the  work  together.  Many  similar  examples 
may  be  done  in  the  same  manner. 

f  ANALYSIS — If  John,  in  2  days,  can  do  the  work,  he  can  do 
\  of  it  in  1  day  ;  and  if  James  and  John  together,  in  \\  days. 
or  -|  days  can  do  the  work,  they  can  do  1  divided  by  f,  that  is,  f 
of  it,  in  1  day  ;  then,  James  can  do  -jj  less  -1  in  a  day;  that  is, 
|  or  ^  of  it ;  hence,  he  can  do  the  whole  in  3  days. 


I,ES.  XIII.]  INTELLECTUAL    ARITHMETIC.  151 

10.  A  can  do  a  piece  of  work  in  3  days,  A  and  B 
together,  can  do  it  in  1|  days :  in  what  time  can  B  do 
it  working  alone  ? 

11.  A  cistern  can  be  filled  by  one  pipe  in  2  hours, 
and  by  2  pipes  in  1 1  hours  :  how  long  will  the  second 
pipe  require  to  fill  it  if  running  alone? 

12.  A  cistern  can  be  filled  by  3  pipes  in  1T\-  hours, 
one  of  the  pipes  can  fill  it  in  2  hours,  and  another  in 
3 :  how  long  will   it  require    the  third  to  fill  it,  if 
running  alone1? 

13.  A  man  and  his  wife  usually  drank  a  gallon  of 
beer  in  12  days  ;  but  when  the  man  was  from   home 
it  lasted  his  wife  30  days  :  how  many  days  would  the 
man  require  to  drink  it] 

14.  A  quantity  of  flour  will   last  one   family  six 
weeks,  and  the  same  flour  will  last  another  family  3 
weeks.     It   is    found    that  one-third  of  the  flour  is 
spoiled :    how    long    will    the   remainder   last    both 
families'? 

15.  If  two  families  consume  a  quantity  of  provi 
sions  in   1^  weeks,  and   one   family  alone  would  con 
sume  the  same  provision  in  4  weeks:  how  long  would 
it  last  the  other  ? 

16.  A  can  mow  a  field  in  1  day,  B  in  2  days,  and 
C  in  3  clays :  in  what  time  can  they  all  mow  it,  work-  ' 
ing  together  1 

17.  A  can  mow  a  field  in  2  days,  B  can  mow  it  in 
3  days,  but  by  the  aid  of  C,  they  can  mow  it  in  j6T  of 
1  day  :  how  long  will  it  take  C  to  mow  it  alone  ? 

In  what  time  can  A  and  B  mow  it? 
In  what  time  can  A  and  C  mow  it? 

18.  Three    carpenters    can    finish    a   house   in   2 
months  ;  two  of  them  can  do  it  in  2-J-  months :  how 
long  will  it  take  the  third  to  do  it  alone? 

19.  A,  with  the  assistance  of  B,  can  build  a  wall  2 
feet  wide  3  feet  high  and  30  feet  long  in  4  days,  but 

^  with  the  assistance  of  C  they  can  do  it  in  2£  days  : 


152  INTELLECTUAL    ARITHMETIC.  [sEC.  VIL 

What  part  of  it  can  A  and  B  build  in  1  day  1 
What  part  of  it  can  they  all  build  in  1  day? 
What  part  of  it  can  C  alone,  build  in  1  day? 
In  how  many  days  would  C  build  it  alone  ? 


LESSON   XIV. 
Analysis  of  Questions. 

1.  A  laborer  engaged  to  work  for  16  days  on  these 
conditions:  For  every  day  he  labored,  he  wras  to  re- 
ceive  4  shillings,  and  for  each  day  that  he  was  idle  he 
was  to  pay  2  shillings  for  his  board  ;  at  the  end   of 
the  time  he  received  52  shillings :  how  many  days 
did  he  work,  and  how  many  days  was  he  idle  ?* 

2.  A  carpenter  took  an  apprentice  on  these  terms : 
he  paid   his  father,  at  the  end  of  each  month  of  20 
working  days,  3   shillings  a  day,  for   every  day  the 
boy   worked — charged   him  20   shillings  for  clothes, 
also,  1  shilling  for  board,  for  every  idle  day  ;  at  the  end 
of  the  time  there  was  30  shillings   due  him  :  how 
many  days  was  he  idle? 

3.  A  merchant  bought  50  yards  of  calico,  some  of 
which  was  damaged,  on  these  terms  :  He  was  to  pay 
3  dimes  a  yard  for  all  that  was  perfect,  and   1   dime 
a  yard  for  all  that  was  injured  ;  at  the  settlement  he 

'paid  12  dollars  :  how  many  yards  were  injured'? 

4.  A  grocer  purchased  30  fowls,  a  part  turkeys  and 
a  part  of  them  chickens ;   for  the  turkeys  he  was  tc 
pay  1 1  dimes  a  piece,  and  for  the  chickens  4  ;  he  paid 
in  all,  24  dollars  and  60  cents  :  how  many  were  there 
of  each  kind? 

*  ANALYSIS. — Had  he  labored  the  16  days,  he  would  have  re 
ceived  the  64  shillings.  But  he  received  only  52  shillings: 
hence,  he  lost  12  shillings  by  idleness.  But  as  he  paid  2  shil 
lings  a  day  for  his  board,  and  lost  4  shillings  a  day  in  wagesN  he 
lost,  in  all,  6  shillings  a  day :  therefore,  the  number  of  days  he 
was  idle  will  be  expressed  by  12  divided  by  6,  giving  2  idle 
days  :  therefore,  he  worked  14  days. 


LES.   XIV.]  INTELLECTUAL    ARITHMETIC.  1 5o 

5.  A  farmer  hired  a  father  and  son  to  do  20  days 
work  between  them  :   the  father  was  to  have  a  dollar 
for  every  day  he  worked,  and   the  son    75   cents  ;  at 
the  end  of  the  time,  the  amout  paid  was  17  dollars  : 
how  many  days  did  each  work  ? 

6.  A  yard  stick  is  broken  into  2  parts,  the  shorter 
of  which  is  f  the  length  of  the  longer  :   what   is  the 
length  of  each  piece  1? 

7.  A  piece  of  cloth  of  40  yards  in   length   is  cut 
into  2  pieces,  such,  that  the  smaller  piece  is  |  of  the 
larger:   What  is  the  length  of  each  piece  ? 

8.  What  number  is  that  to  which  if  |  of  itself  be 
added,  the  sum  will  be  32  ? 

9.  A  coat  and  vest  cost  24  dollars,  and   the  vest 
cost  i  as   much  as   the   coat:   what  was  the  cost  of 
each? 

10.  A  cow  and  calf  are  worth  56  dollars,  and  the 
calf  is  worth  ^T  of  the  cow  :  what  is  the  value  of 
each? 

11.  A  pole  16  feet  long  stands  in   the  mud,  water, 
and  air.     The  part  in  the  mud  is  ^  of  the  part  in  the 
water,  and  the  part  in   the  ,air  is  equal  to  the  other 
two  :   what  is  the  length  of  each  part? 

12.  There  is  a  fish  weighing  72  pounds.     His  head 
weighs  twice  as  much  as  his  tail,  and  his  body  weighs 
as  much  as  his   head  and  tail  together  :  what  is  the 
weight  of  each  part  ? 

lo.  Divide  the  number  52  into  two  such  parts, 
that  |  of  the  larger  part  shall  be  equal  to  the  less 
part. 

*  ANALYSIS. — The  fractional  unit,  in  this  question,  is  one  fifth 
the  length  of  the  longer  piece.  There  are  5  of  these  unite  in  the 
longer  piece  and  4  in  the  shorter :  hence,  there  are  9  in  the 
whole  stick,  which  is  36  inches  long.  The  value  of  the  frac 
tional  unit,  is,  therefore,  36  divided  by  9,  or  4  inches.  Hence, 
one  of  the  pieces  is  20  inches,  and  the  other  16.  All  simi/ar 
questions  are  solved  by  finding  the  fractional  unit. 


154  INTELLECTUAL    ARITHMETIC.  [SEC.  VTL 

14.  A  tailor  has  48  yards  of  cloth  in  2  pieces;  | 
of  the  longer  piece  is  equal  to  |  of  the  shorter:   how 
many  yards  in  each  piece  1* 

15.  A  farmer  bought  pigs  and  sheep,  in  all  33  :  | 
of  the  pigs  was  equal   to  ^  of  the  sheep  :   how  many 
were  there  of  each  kind  ? 

10.  Three  fourths  of  a  son's  age  is   equal   to  on 
fourth  the  age  of  the  father,  and  the  sum  of  their  ages 
is  80  years  :   what  is  the  age  of  each? 

17.  There  are   125  sheep   in    two  fields,  |    of  the 
number  in  one  field  being  equal  l£  times  the  number 
in  the  other:  how  many  in  each  h'dd  ? 

18.  A  man  after  counting  his  gains  at  play,  found 
that  he  had  increased  his  money  by  \    of   £,  and  that 
he  then  had  42  dollars  :  how  much  had  he  at  first? 

10.  The  difference  of  two  numbers  is  G,  and  the 
less  number  is  £  of  the  greater  :  what  are  the  num 
bers  ?f 

20.  A  father's  age  is  such,  that  f  of  it  is  equal  to 
If  the  age  of  his  son,  and  the  difference  of  their  ages 
is  30  years  :  what  is  the  age  of  each  ? 

21.  What  number  is  that  which  being   diminished 
by  the  difference  between  f  and  f  of  itself,  leaves  a 
remainder  equal  to  34  1 

22.  A  flag  staff  52  feet  long,  is  so  broken  by  the 
wind,  that  ^  of  the  top  piece  is  equal  to  f  of  the  piece 
left  standing  :   how  long  are  the  pieces  ? 

*  SUGGESTION. — If  f  of  the  shorter  piece  equals  f  of  the 
longer1,  then  j  of  the  shorter  equals  \$  of  the  longer,  and  the 
whole  of  the  shorter  piece  equals  y^  of  the  longer.  Always 
find  tJie  unit  of  the  smaller  lit  terms  of  t lie  larger  number. 

f  ANALY§IS. — Suppose  the  greater  number  be  divided  into  8 
equal  parts  ;  two  such  parts,  are  found  in  the  lesser  number: 
tience,  the  difference  between  the  numbers  is  equal  to  J-  of  the 
greater,  which  is  6  :  therefore,  the  greater  number  .&  *8  and  the 
less  12. 


LES.     £V.]  INTELLECTUAL    ARITHMETIC.  155 

23.  James  was  asked  how  many  marbles  he  had, 
and  replied,  I  have  55,  black   and  red,  and  |  of  the 
black  make  just  as  many  as  |  of  the  red.     Pray  how 
many  have  I  of  each  sort  1 

24.  A  laborer  engaged  to  work  20  days,  and  was 
to  receive  9  shillings  for  every  day  he  worked,  and 
pay  3  shillings  a  day  for  his  board,  every  day  that  he 
was  idle ;  in  settling,  he  received  84  shillings :  how 
many  days  did  he  work  ? 

25.  A  church,  including  the  steeple,   is   188  feet 
high.     If  the  height  of  the  steeple  is  equal  to  $  the 
height  of  the  body  of  the  building,  vhat  is  the  height 
of  each  1 


LESSON  XV. 
Analysis  of  Questions. 

1.  William  chases  Henry,  who   is  42  feet  in  ad 
vahce,  around  a  circular  walk   of  100  feet.     Their 
steps  are  each  3  feet,  but  William  takes  6  steps  while 
Henry  takes  but  5:  how  many  steps  must  William 
make  to  overtake  Henry  ? 

2.  A  hare  is  25  of  his  own  leaps  before  a  grey 
hound,  which  is  pursuing  hrm.     The  grey  hound  makes 
2  leaps  while  the  hare  makes  5;  but   1   leap  of  the 
greyhound  is  equal   to  3  of  the  hare's  :  how  many 
leaps  will  the  greyhound  make  before  he  overtakes 
the  hare  ?* 

3.  James  is  in  pursuit  of  John,  and    12  of  John's 
steps  behind   him.     James  steps  3  times  while  John 


*  ANALYSIS. — Since  the  greyhound  makes  2  leaps  while  the 
hare  makes  5,  in  the  time  that  the  greyhound  makes  1  leap 
the  hare  will  make  2£  leaps.  But  1  leap  of  the  greyhound  is 
equal  to  3  leaps  of  the  hare ;  hence,  every  time  the  greyhound 
jumps  he  will  gain  on  the  hare  £  of  a  hare's  leap  :  therefore,  he 
must  make  50  leaps  to  overtake  him. 


15(>  INTELLECTUAL    ARITHMETIC.  [SEC.  VII 

steps  4  times  ;  but  James'  steps  are  twice  as  long  as 
John's :  how  many  steps  must  Jarnes  make  to  over 
take  him1? 

4.  Mary  is  24  years  old  and  Jane  is  4:  howirany 
years   must  elapse  before  Mary's  age   will   be  iust 
double  Jane's  ]* 

5.  A  father  is  60  years  old  and  his  son  is  35 :  how 
long  since  the  age  of  the  father  was  double  that  of  the 
son  ?     What  was  then  the  age  of  each  1 

6.  A  mother  is  36  years  old  and  her  daughter  12  : 
how  long  before  the  age  of  the  mother  will  be  double 
that  of  the  daughter  ?     What  will  then  be  the  age  of 
each  ? 

7.  A  mother  is  48  years  old  and  her  daughter  30  : 
how  long  since  the  age  of  the  daughter  was  half  that 
of  the  mother  ?      What  was  then  the  age  of  each  ? 

8*  A  mother  is  48  years  old  and  her  daughter  10  : 
how  long  before  the  age  of  the  daughter  will  be  one- 
third  that  of  the  mother  ?f  What  will  then  be  the 
age  of  each  ? 

9.  A  mother  is  54  years  old  and  her  daughter  20 : 
how  long  since  the  age  of  the  daughter  was  one-third 
that  of  the  mother  ?  What  was  then  the  age  of  each  1 

*  ANALYSIS. — At  Jane's  birth,'4  years  ago,  Mary  was  20  years 
old,  and  Jane's  age  was  0.  Twenty  years  from  that  date,  Mary 
will  be  twice  as  old  as  Jane  :  Hence,  if  Mary1  sage  be  diminished 
by  Jane's,  and  the  remainder  be  multiplied  by  2,  the  product  will 
denote  Mary's  age  when  it  is  double  Jane's. 

f  ANALYSIS. — At  the  birth  of  the  daughter,  the  mother  was  3 8 
years  old.  What  number  added  to  38  years  will  give  a  sum 
equal  to  3  times  the  number  added  ? 

If  3  times  the  number  added  is  equal  to  38  plus  the  nunv 
her,  then,  twice  the  number  added  will  be  equal  to  38  ;  and 
the  number  must  be  one  half  of  38,  or  19  years.  Hence,  if  the 
daughter's  age  be  subtracted  from  the  mother's,  and  the  remainder 
divided  by  2,  the  quotient  will  be  the  daughter's  age,  when  it  is 
one-third  the  mothers.  If  the  difference  be  divided  by  3,  the 
quotient  will  be  the''  daughter's  age  when  it  is  one-fouth  the 
mother's  ;  and  so  on. 


LES.   XVI.]  INTELLECTUAL    ARITHMETIC.  157 

10.  A  father's  age  is  45  years,  and  his  son  9  :  how 
long  before  the  age'  of  the  son  will  be  one  fourth  that 
of  his  father?     What  will  then  be  the  age  of  each? 

11.  A  father  is  54  years  old,  and   his  son  30  :  how 
long  since  the  age  of  the   father  was  4  times  that  of 
Jio  son  1     What  was  then  the  age  of  each  1 


LESSON    XVI. 

Analysis  of  Questions  by  means  of  Unity. 

WE  have  said,  that  any  number,  regarded  as  a 
whole,  may  be  called  UNITY. 

What  may  any  number  be  called,  when  it  is  re 
garded  as  a  whole? 

1.  What  number  added  to  twice  itself  will  give  a 
sum  equal  to  12  1* 

2.  What    number    is    that   which. added   to  three- 
times  itself  will  give  a  sum  equal  to  24? 

3.  What  number  is  that  which  added  to  half  itself 
will  give  6  ? 

4.  What  number  added  to  half  itself  will  give  9? 

5.  What  number  added  to  one-fourth  of  itself  will 
give  20  ? 

6.  Wrhat  number  is  that  which  added  to  twice  itself, 
and  the  sum  to  3  times  itself  will  give  30  ? 

7.  What  number  ad<Jed  to  half  of  itself,  and  to  one 
fourth  of  itself  will  give  28  ? 

8.  James  being  asked  his  age,  said,  I  am  half  the 
age  of  my  father,  and  the  sum  of  our  ages  is  60  years  : 
What  is  the  age  of  each  ? 

*  ANALYSIS. — Call  the  number  sought,  unity.  Then  by  the 
conditions  of  the  question,  unity  plus  twice  unity  is  equal  to  12. 
But  unity  plus  twice  unity  is  equal  to  3  times  unity,  which  is 
equal  to  12.  Then,  if  3  times  unity  is  equal  to  12,  once  unity  is 
equal  to  12  divided  by  3,  which  is  4. 

SUGGESTION. — Let  the  pupil  see  if  4  will  fulfil  the  conditions : 
4-1-4  X  2  ~~  4  +  8  —  12.     Let  every  question  of  the  lesson  lx/ 
analyzed  and  verified  in  a  similar  way 
14 


158  INTELLECTUAL    ARITHMETIC.  [SEC.  VII, 

9.  What  number  is  that  to  which  if  its  £  be  added, 
llie  sum  will  be  35  ? 

10.  A  man  being  asked  his  age,  replied,  if  to  my 
age  you  add  one-third  of  it  and  then  £  of  it,  the  sum 
will  be  57  :   IIo\v  old  was  he  ? 

11.  John  being  asked  how  many  marbles  he  had, 
aid,  that  one  half  of  what  he  had,  increased   by   ^t 

and  diminished  by  {,  was  equal  to  14. 

1'2.  Divide  the  number  1*2  in  two  such  parts,  that 
one  shall  be  three  times  the  other? 

13.  Divide  24  into  3  such  parts,  that   the  second 
shall  be  3  times  the  first,  and  the  third  4  times   the 
first :   What  are  the  numbers  ? 

14.  Divide  32  into   3  such  parts,  that  the  second 
shall  be  4  times  the  first,  and  the  third  3  times  the 
first:   What;  are  the  numbers  ? 

15.  Divide  27  into  two  such  parts,  that  the  second 
shall  be  f  of  the  first? 

10.  Divide  20  into  three  such  parts,  that  the  second 
shall  be  one  half  of  the  first,  and  the  third,  one  third 
of  the  second  ? 

17.  James   asked   Robert  how  many   marbles  he 
had.     Robert  replied  :  "  If  you  will  give  me  one-half 
the  number  I  now  have,  and  then  one-third  of  that 
number,  and  William  will   then  give  10,  1  shall  have 
one  hundred  and  twenty  :"    how  many  had  he  ? 

18.  A  young  chap  asked  an  old  gentleman  his  age, 
who  replied,  u  When  you  were  born,  I  was  f  of  my 
present  age  :  one  third  of  your  age  plus  one  fourth  of 
it,  is  equal  to  7  years.     Can  you  now  tell  how  old  I 
am"?" 

19.  A  tailor  buys  a  piece  of  cloth  for  6  dollars  a 
yard,  and  a  piece  of  equal  length  for  $2  a  yard,  and 
sells  the  whole  at  $4  a  yard  :  does  he  make  or  lose? 

20.  John  and  James  together  have  45  marbles :  if 
John's  is  equal  to  ^  of  James,  how  many  has  each  1 


LES.  XVII. J  INTELLECTUAL    ARITHMETIC.  159 

21 .  A  market  woman  bought  geese  and  turkeys  ;  36 
in  nil  ;  one  seventh  of  one  sort  was  equal  to  one  half 
of  the  other:  how  many  of  each  kind  ? 

2*2.  A  pole  72  feet  long  has  one  half  as  much  in 
the  mud  as  in  the  water,  and  twice  as  much  in  the 
air  as  in  the  mud  and  water  together  :  how  many  feet 
in  each  1 

23.  What  number  added  to  -J-,  to  J,  and  to  ^  of 
itself  will  give  a  sum  equal  to  50? 

24.  Mr.  Wilson  bought  a  hat,  a  coat,  and  a  cloak  ; 
he  paid  for  the  coat  1^-  as  much  as  for  the   hat,  and 
for  the  cloak, three  times  as  much  as  for  the  coat ;  and 
for  all  he  paid  42  dollars;  how  much  did  he  pay  for 
each] 


LESSON  XVII.  v* 

Analysis  of  Questions  by  means  of  Unity. 

1.  What  number  is  that  to  which  if  10  be  added, 
the  sum  will  be  16?* 

2.  James  being  asked  how  many  marbles  he  has. 
replied,  if  to  -J  of  what  1  have  you  add  8,  the  sum  wiL 
.be  equal  to  18:  how  many  had  he] 

3.  What  number  is  that  whose  half  exceeds  it* 
third  by  4? 

4.  What  number  is  that,  to  which  if  £  of  itself  and 
4  be  added,  the  sum  will  be  equal  to  22? 

5.  What  number  is  that,  which  being  added  to  on* 
third  of  itself,  and  to  3  times  itself,  will  give  a  SUIT 
equal  to  26  ? 

6.  What  number  is  that,  which  being  diminished 
by  its  half  and  its  third,  the  remainder  will  be  equa 
to  4? 


*  ANALYSIS. — Denote  the  required  number  by  unity.  Theiy 
by  the  conditions  of  the  question,  unity  plus  10  equals  1 6.  But  if 
unity  plus  lOequalb  16,  unity  must  be  equal  to  16  minus  10,  or  I 

For,  if  the  same  number  be  subtracted  from  two  equal  r.uir  ••* 
the  remainders  will  be  equal. 


160  INTELLECTUAL    ARITHMETIC.  [SEC,  VH, 

7.  What   number   is   that,    to   the   one   fourth   of 
which  if  10  be  added,  the  sum  will  be  equal  to  20  ? 

8.  What  number  is  that  to  which  if  its  one  fourth 
be  added,  and  the  sum  diminished  by  7  will  leave  13 
for  a  remainder  ? 

9.  A  tailor  cuts  a  piece  of  cloth  21   yards  long 
into   3   pieces;    the   second    contains  \    of  the    first; 
minus  5  yards,  and  the  third  is  equal  to  \  of  the  first 
plus  5  yards:  how  many   yards    are  there  in  each 
piece  1* 

10.  One  fourth  of  William's!  age  is  equal  to  one 
half  of  John's,  and  the  sum  of  their  ages  is  24:  what 
is  the  age  of  each  ? 

•  1 1.  If  one  half  of  Charles'  age  equals  one  sixth  of 
John's,  and  the  sum  of  their  ages  is  16,  what  is  the 
age  of  each  ? 

12.  Divide  15  into  two  such  parts,  that  one  eleventh 
of  the  first  shall  be  equal  to  \  of  the  second. 

13.  A  watch  and  seal  are  together  worth  64  dollars, 
and  the  watch  is  worth  7  times  as  much  as  the  seal : 
what  is  the  value  of  each  1 

14.  A  mother  divided  56  pins  between  Jane  and 
Nancy,  so  that  one  fifth  of  Jane's  was  equal  to  one 
half  of  Nancy's  :  how  many  had  each  ? 

*  ANALYSIS. — Denote  the  first  part  by  unity.     Then 

unity  =  the  first  part, 
-|  of  unity  —  5  =  the  second  part,  and 
|  of  unity  -f-  5  =  the  third  part ;  and  since 
tlia  sum  of  the  parts  is  equal  to  21  yards, 

unity  -f  i  unity  —  5  -f  £  of  unity  +  5=21;  that  is, 
J  of  unity  =  21  ;  and 
J  of  unity  =  3,  or  unity  =  12. 
Hence,  the  pieces  are  12,  1,  and  8  yards. 


LE8.   XVII.]          INTELLECTUAL    ARITHMETIC.  161 

15.  John  gave  one  third  of  his  marbles  to  William, 
and  then  gave  half  of  what  he  had  left  and  4  more  to 
Charles,  after  which  he  had  8  remaining:  how  many 
had  he  at  first? 

16.  James  and  John  together,  have  50  marbles  ;  3 
times  James'  number  is  equal  to  7  times  John's: 
how  many  has  each  1 

17.  The  sum  of  two  numbers  is  16,  and  the  greater 
is  3  times  the  smaller':  what  are  the  numbers? 

18.  Two  partners  in  trade  have  made  a  profit  of 
99  dollars,  and  agree  to  divide  it  so  that  the  second 
shall  have  4  dollars  every  time  the  other  has  5  :  what 
was  the  portion  of  each  1 

19.  A  school  of  88  scholars  has  three  classes;  the 
second  contains  1-J-  times  as  many  as  the  first,  and  the 
third    twice   as   many    as   the    second :    how    many 
scholars  in  each  class? 

20.  If  ^  of  John's  marbles  is  equal  to  |-  of  James', 
and  together  they  have  56,  how  many  has  each? 

21.  A  piece  of  cloth  is  divided  into  3  parts;  one 
piece   is  4  yards   long,  which  is  one  eighth  of  the 
length  of  the  other  two,  but  of  these  two  pieces  the 
longer  is  3  times  the  shorter :  what  is  the  length  of 
each  piece? 

22.  Two  persons,  A  and  B,  at  a  tavern,  spend  80 
cents,  of  which  twice  what  A  spends  is  equal  to  3 
times  what  B  spends  :  how  much  is  spent  by  each? 

23.  The  sum  of  the  ages  of  two  persons  is  56  years, 
and  twice  the  age  of  the  elder  is  6  times  the  age  of 
the  younger :  what  is  the  age  of  each  ? 

24.  A  man  has  66  fowls,  and  after  selling  a  part  of 
them    found    that  what   he  had  left  was  twice  the 
number  sold:  how  many  did  he  sell,  and  how  many 
had  he  left  ? 

25.  A  man  after  spending  a  part  of  his  money  at  i 
tavern,  found  that  what  he  had  left  was  one  fifth  < 

14* 


162  INTELLECTUAL    ARITHMETIC.  [sEC.   VII. 

what  he  had  spent,  and  remembering  that  he  had  84 
cents  at  first,  wished  to  know  how  much  he  had  left  ? 

26.  A  man  sold  a  horse  for  140  dollars,  by  which 
he  gained  |-  of  what  the  horse  cost  him :  what  did  he 
give  for  the  horse  ? 

27.  John  and  Charles  receives  18  cents  for  premiums, 
t  school.     If  three  times  John's  money  be  subtracted 

from  3  times  what  both  receive,  the  remainder  will 
be  24  :  how  much  does  each  receive  1 

28.  A  fish  weighs  64  pounds.     The  head  weighs  3 
times  as  much  as  the  tail,  and  the  body   weighs  as 
much  as  the  head  and  tail  both :  what  is  the  weight 
of  each  part  ? 

29.  If  a  shadow  15  feet  long  is  cast  by  a  stick  10 
feet  long,  standing  vertically,  what  will  be  the  length 
of  a  stick  or  pole,  like  placed,  which  casts  a  shadow 
24  feet  long,  at  the  same  time  of  day  1 

30.  If  a  shadow  10  feet  long  is  cast  by  a  stick  5 
feet  high,  what  will  be  the  length  of  a  stick,  standing 
in  a  like  position,  which  casts  a  shadow   16  feet  in 
length,  at  the  same  time  of  day  ? 

31.  A  tailor  cut  6  coats  from  a  piece  of  cloth,  after 
which  it  measured  24  yards ;  he   then  cut  5  pairs  of 
pantaloons,  which  took  -|-  as  much  as  the  coats,  when 
it  was  found  that  one  half  of  the  piece  was  left :  how 
many  yards  did  the  piece  contain  ? 

32.  A  farmer  has  two  bins  for  grain,  which  together, 
hold  60  bushels  ;  if  he  diminishes  the  greater  by  ^  of 
its  capacity  and  increases  the  less  by  the  same  amouni 
the  two  will  then  hold  an  equal  number  of  bushels  : 
how  many  bushels  does  each  hold  ] 


LES.  XVIII.]        INTELLECTUAL    ARITHMETIC.  103 

LESSON  XVIII. 
Promiscuous  Questions. 

1.  A  man  sold  a  barrel  of  flour  for  $4,  which  was 
|  of  what  it  cost :  how  much  did  he  lose  ? 

2.  If  |  of  a  piece  of  broadcloth,  containing  18  yards, 
cost  32  dollars,  how  much  was  that  a  yard  ] 

3.  A  man  sold  a  cow  for   18  dollars,  which  was  | 
of  what  she  was  worth  :  what  was  her  true  value  1 

4.  A   man  spends  -J-  of  his  monthly  income  for  a 
hat,  and  twice  as  much  for  a  coat,  and  has  $10  left : 
how  much  does  he  receive  a  month  1 

5.  A  man  pays  J  of  his  daily  wages  for  board,  ar.d 
%  for  his  clothes,  and   at  the  end  of  the  week  has 
saved  81^  '  what  are  his  wages  a  day  ] 

6.  A  pole  stands  £  in  the  air,  £  in  the  water,  and  3 
feet  in  the  mud  :  how  long  is  it  ? 

7.  The  body^f  a  fish  is  three  times  as  long  as  his 
head,  and  his  tail   is  2  feet,  which  is  one  fourth  the 
length  of  the  head  and  body  :   what  is  the  length  of 
each  part,  and  of  the  entire  fish  ] 

8.  A  person  gave  -J-  of  his  money  to  each  of  5 
persons,  and  had  4  cents  left :  how  much  had  he  at 
first? 

9.  John  gave  Charles  4  times  as  many  apples  as 
he  gave  to  William,  and  to  William  i  as  many  as 
he  had  left,  which  was   ten :    how  many  had  he  at 
first  ] 

10.  What  will  be  the  cost  of  a  bag  of  coffee,  if  f  of 
it  cost  $4f  1 

11.  Mary  gave  f  of  her  money  to  Jane,  and  J  to 
Eliza,  and  had   3  cents  remaining  :  how  much   had 
she  at  first  ] 

12.  If  3  men  can  do  a  piece  of  work  in  8  days, 
how  long  will  it  take  12  men  to  do  the  same  work  ] 

13.  If  5  men  can  do  a  piece  of  work   in  7£  days, 
how  long  will  it  take  nine  men  to  do  the  same  work] 
How  long  12  men]     18  men]     30  me/.i]     72  men? 


104  INTELLECTUAL    ARITHMETIC.  [SEC.  VII. 

14.  A  man  would  give  -J-,  ^,  and  J-  of  the  money  in 
his  purse  to  the  oldest,  the  second,  and  youngest  son; 
now.  if  he  has  fifty-two  dollars  left,  how  much  will 
each  receive? 

15.  James  and  John,  on  comparing  their  marbles, 
find  that  together,  they  have  112;  and  that  if  James 
givos  J  of  his  to  John,  they  will  have  an  equal   num 
ber  •   how  many  has  each  1 

16.  James,  John,  and  Charles  together,  have  120 
marbles;   if  James  gives  £  of  his  to  John,  each  will 
hnve  4-  as  many  as  Charles  :  how  many  marbles  has 
each  ?* 

17.  James,  John,  and  Charles  together,  have  800 
marbles;   if  James  gives  12  more  than  |  of  his  to 
John,  each   will  have  J   as  many   as  Charles  :  how 
many  has  each  ] 

18.  In  a  fruit  orchard,  one  third  ofrall  the  trees  are 
apples.  ^  are  pears,  £  are  cherries,  and  26  are  plums : 
how  many  of  each  sort,  and  how  many  in  all] 

19.  A   person   being   asked  the  time  of  clay,  said, 
that  the  time  past  12  o'clock  M.  (that  is  noon),  was  -J 
the  time  past  the  previous  midnight:  what  was  the 
time] 

How  many  equal  parts  (fourths),  from  12,  mid 
night,  to  the  required  time]  How  many  of  these 
parts  between  midnight  and  noon]  What  is  the 
value  of  each  part  ? 

20.  How  will  you  divide  half  a  water-melon  among 
3  boys,  so  that  the  second  shall  have  twice  as  much 
as  the  fivst,  and  the  third  twice  as  much   as  the  sec 
ond  :    what  part  of  the  whole  melon   will   each  boy 
have  ? 

Koi  E. — A  fter  James  has  given  one  sixth  of  his  marbles  to  John, 
each  will  have  one  fourth  of  120,  or  30,  and  Charles  will  have 
one  half  of  120  or  60.  Then,  what  number  diminished  by  £ 
leaves  80  ? 


LES.  XIV.]  INTELLECTUAL    ARITHMETIC.  105 

21.  If  f  of  a  barrel  of  flour  will  last  a  family  of  5 
persons  10  days,  how  long  will  a  barrel  last  a  family 
of  25  persons'? 

22.  If  5|  baskets  of  peaches  are  worth  $8$,  how 
many  potatoes  at  25  cents  a  bushel,  will  one  basket, 
buy? 

23.  If  25  bushels  of  oats  will  serve  5  horses  for  7 
days,  how  many  will  serve  7  horses  for  9  days? 

24.  A  farmer  has  his  sheep  in  3  partures.     In  the 
first  he  has  ±  of  his  flock,  in  the  second  £,  in  the  third 
J,  and  G  over:  how  many  has  he  in  all  ? 

25.  A  tailor  has  a  piece  of  cloth,  from  which  he  cuts 
enough  for  a  suit  of  clothes,  and   finds  that  he  has  28 
yards  left,  which  is  just  ^  of  f   of  his  piece:   how 
many  yards  in  the  piece? 

20.  A  man  starts  on  foot  from  Albany,  for  New 
York,  and  at  the  end  of  the  first  day  finds  that  he  has 
125  miles  yet  to  travel,  which  \vas  just  -|  of  the  whole 
distance  :  what  was  the  whole  distance? 

27.  A  grocer  bought  an  equal  number  of  lemons 
and   oranges ;  he  paid  9  cents  for  every  2  oranges, 
and  7  cents  for  every  4  lemons  :  what  must  he  sell 
them  at  a  piece  to  make  100  percent? 

28.  A  grocer  bought  a  certain  number  of  eggs  at 
the  rate  of  2  for  5  cents,  and  an  equal  number  at  the 
rate  of  3  for  7  cents,  and  sold  them  at  Scents  a  piece, 
by  which  he  made  21  cents  :  how  many  eggs  did  he 
buy  ? 

29.  A  man  and  his  wife  consumed  ten   pounds  of 
meat  in  3  days.     The  man  alone  would  have  con- 
sumed  it  in  5  days:  what  part  of  the  meat  did  the 
v\oman  consume? 

30.  If  two  men  can  dig  32   bushels  of  potatoes 
in  1  day,  working  8  hours  a  day,  how   long  will  it 
take    3    men,  working   9  hours   a   day,  to    dig    54 
bushels  ? 


166  INTELLECTUAL    ARITHMETIC.  [SEC.  VII. 

31.  If  5  men  can  build -a  fence  20   rods  long  in  8 
days,  how  long  will  it  require  16  men  to  build  it? 

32.  What  nuftiber  is  that  from  which  if  2  be  sub 
tracted  §  of  the  remainder  will   be  4?     What  num 
ber  is  that  f  of  which   is  4  ?     If  2  be  added,  what 
number? 

33.  After  paying  away  ^  and  |-  of  my  money,  I 
had  10  dollars  remaining:  how  much  had  1  at  first? 

34.  A  person  after  spending  i  of  his  money,  and 
then  jf  of  the  remainder,  had   8  dollars    left :  how 
much  had  he  at  first  1 

35.  A  man  at  play   lost  £  of  his  money,  and  the 
next  night  lost  ^  of  the  remainder,   when   he  found 
that  he  had  but  $12   left:  how  much  had  he  at  first 7 

36.  John  is  20  paces  ahead  of  Charles,  but  Charles 
takes  2  steps  while  John   takes  1  :    how   many   steps 
will  John  make  before  he  is  overtaken?     How  much 
does  Charles  gain  at  each  step1? 

37.  If  John  is   15  steps  in   advance,  and  Charles 
makes  3  steps  while  John  makes  2 :  how  many  steps 
will  Charles  make  before  he  comes  up? 

38.  John  being  asked  how  many  marbles  he  had. 
said,  1  have  but  four,  for  James  first  took~  £  of  all  I 
had,  and  Charles  then  took  ^  of  what  was  left :  how 
many  had  he  before  he  lost  any  ? 

39.  A  man  bought  a  harness,  a  carriage,  and  a  pair 
of  horses.     The  horses  cost  70  dollars  more  than  the 
carriage,  and  the  carriage  50  dollars  more  than  the 
harness,  which  cost  40  dollars  :   what  was  the  whole 
cost  ? 

40.  A  tailor  cut  a  piece  of  cloth  into  3  parts  :  the 
first  part  contained  2  yards  more  than  the  second; 
the  second  contained  4  yards  less  than  the  third  ;  and 
the  third  was  found  to  contain   12  yards:  how  many 
yards  in  the  piece? 

41.  A  tailor  cut  a  piece  of  cloth  into  3  parts:  *^ 


LES.  XIV.]  INTELLECTUAL    ARITHMETIC.  IQ1 

ftrst  part  contained  -J-  of  the  piece  and  4  yards  over ; 
the  second  contained  i  ®f  the  piece  wanting  2  yards; 
and  the  third  contained  J  of  the  piece  and  6  yards 
over :  how  many  yards  in  the  whole  piece,  and  how 
many  in  each  part? 

42.  A  man  paid  f  of  his  year's  bill ;  after  which  he 
paid  i  of  what  was  left,  and  yet  owed  12  dollars : 
hosv  much  was  the  bill? 

43.  A  man    spends   in  a  pleasure  trip  J   of   his 
money  at  a  hotel,  -^  of  it  in  a  railroad  fare,  and  14 
dollars  in  carriage  hire   and   other    expenses :    how 
much  did  he  spend  in  all  ? 

44.  Mr.  Wilson  spends  |  of  his  money  at  a  tavern, 
and  then  goes  to  a  grocery  and  pays  f  of  the  remain 
der  for  coffee  and  tea ;  he  counted  what  he  had  left, 
and  found  2  dollars:  how  much  had  he  at  first  ? 

4-5.  A  young  lady  had  a  portion  at  marriage.  She 
expended  |-  of  it  in  furniture,  gave  6  dollars  to  each 
of  two  sisters,  and  had  28  dollars  left :  how  much  had 
she  at  first? 

46.  James  has  half  as  many  marbles  as  Charles 
and  10  over ;  William  has  half  as  many   as  James, 
and  Charles  has  as  many  as  James  and  William  :  how 
many  has  each? 

47.  A  tailor  wishes  to  divide  a  piece  of  cloth  con 
taining  50  yards,  into  two  such  parts  that  one  part 
shall  be  5£  times  the  other  :  how  many  yards  in  each 
piece  ? 

48.  A  farmer  has  a  field  containing  30  acres  of 
land,  and  wishes  to  divide  it  into  3  such  fields   that 
the  second  shall  be  double  the   first,  and  the   third 
equal  to  the  sum  of  the  other   two  :  how   much  in 
each  of  the  new  fields  ? 

49.  A  father  buys  70  marbles  for  John  and  Henry , 
and  wishes  that  John  should  have  2-J-  times  as 

as  Henry  :  how  must  he  divide  them  ? 


1GS  INTELLECTUAL    ARITHMETIC.  [SEC.  VH. 

50.  A    tailor's    bill    amounts   to    96   dollars;    4 
articles  were  charged,  vests,  pantaloons,  and  coats; 
(here  was  3  times  as  much  charged  for  pantaloons 
as  for  vests,  and  twice  as  much  for  coats  as  for  vests 
and  pantaloons  :  how  much  was  charged  for  each  kind? 

51.  A  tailor  having  a  piece  of  cloth,  cut  it  into 
two  parts,  one  of  which  was  2  yards  less  than  one 
half  the  piece ;  he  then  cut.  the  smaller  piece  into  two 
equal  parts,  when  he  found  that  each  part  contained  14 
yards :  how  many  yards  in  the  piece  ? 

52.  A    mason  built  a  wall  in  2  days.     The  first 
day  he  built  ^  of  it  and  2  rods  over;  the  second  day 
he  built  -|  of  the  remainder  and  3  rods  more :  what 
was  the  length  of  the  wall  ? 

53.  A  horse  is  worth  41   times  the  saddle,    and 
both  are   worth   110  dollars:  what  is  the  value  of 
each? 

54.  A  father  divided  a  farm  of  130  acres  between 
8  sons;  the  second  was  to  have  1^  times  as  much  as 
the  first,   and  the  third  to  have  £  as  much  as  the 
second  :  what  was  the  share  of  each  ? 

55.  A  merchant  in  settling  up  his  cash  account, 
found  that  if  he  had  £  and  |  more,  that  he  would  still 
need  4  dollars  to  make  §70:  how  much  had  he? 

50.  The  difference  between  |  and  |  of  a  number  is 
8  less  than  -g-  of  the  number  :  what  is  the  number  ? 

57.  The  difference  between  -  and  f  of  a  number  is 
5  greater  than  ^  of  the  number  :  what  is  the  number  1 

58.  A  man  on  foot  is  35  miles  in  advance  of  a 
man  pursuing  him  on  horseback  ;  the  footman  travels 
i*J  miles  an   hour,  and  the  horseman  rides  7 :  how 
U  ng  before  the  footman  wrill  be  overtaken  ? 

59.  A   dog    pursues    a  fox,  which   is  10  rods    in 
advance;  while  the  fox  runs  3  rods,  the  dog  runs  5: 
how  many  rods  will  the  dog  run  before  overtaking 
the  fox  ? 


LES.  XVIII.]         INTELLECTUAL    ARITHMETIC.  169 

CO.  A  cistern  is  filled  by  a  pipe  which  runs  5  gal 
lons  a  minute;  while  the  water  is  discharged  by  a 
leak,  at  the  rate  of  2  gallons  a  minute :  if  the  cistern 
holds  120  gallons,  haw  long  will  it  take  to  fill  it,  and 
how  many  gallons  will  have  run  out? 

61.  The  minute  hand  of  a  clock  moves  12  times 
as  fast  as  the  hour  hand,  and  moves  over  one  space 
on  the  face,  in  five  minutes:  how  long  will  it  take 
the  minute  hand  to  overtake  the  hour  hand  when   it 
is  one  space  behind  ? 

62.  A  poultry-yard  contains  geese,  ducks,  and  15 
turkeys  ;  if  there  were  10  more  ducks  the  number 
would  be  equal  to  that  of  the  geese  and  turkeys  ;  and 
the  number  of  the  geese  is  equal  to  |  the  number  of 
the  ducks  :  how  many  are  there  of  each  sort  ? 

63.  A  father  distributed  a  sum  of  money  between 
his  three  sons,  thus  :  to  John  he  gave  f-  of  the  whole 
and  9  dollars  over;  to  Reuben  he  gave  15  dollars; 
and  to  William  he  gave  the  remainder,  which  was  | 
of   the    sum     that  he  gave  to  his  other  two  sons; 
bow  much  money  did  he  distribute? 

64.  A  piece  of  cloth  containing  86  yards,  is  cut 
into  two  parts :  |  of  the  whole  piece  is  equal  to  f  of 
the  smaller:  how  many  yards  must  be  added  to  the 
less  piece  to  make  the  two  pieces  equal  ? 

65.  A  person  being  asked  his  age,  replied  :  |-  of 
my  age  added  to  20  equals  my  age  diminished  by  5. 

66.  A  man  bought  a  horse  and  a  colt;  2£  times 
what  he  paid  for  the  colt  equalled  1^  times  50  dollars, 
which  he  paid  for  the  horse  :  what  did  he  pay  for  the 
colt,  and  how  much  more  for  the  horse  than  the  colt? 

67.  A  farmer  bought  a  cow  and  a  sheep,  and  paid 
40  dollars  for  both.     He  paid  for  the  sheep  one  half 
what  lie  paid  for  the  cow,  less  8  dollars :  what  did  he 
pay  for  each  1 

15 


170  INTELLECTUAL    ARITHMETIC.  [SEC.  VII. 

68.  James  has  8  dollars  more  money  than  John ; 
5^  times  this  difference  equals  1|-  times  James'  :  how 
much  has  each? 

GO.  Charles  after  eating  \  of  his  chestnuts  gave 
away  £  of  what  he  had  left,  and  then  had  16  remain 
ing :  how  many  had  he  at  first? 

70.  A  and  B  enter  into  trade  together;  A  puts  in 
2  dollars  every  time  B  puts  in  5 :  A's  money  re 
mains  in  12  months  and  B's  8  ;  they  make  a  profit  of 
128  dollars :  how  should  it  he  divided  between  them  ? 

71.  James  went  out  hunting,  and  shot  one  of  every 
5  squirrels  which  he  saw;  had  he  seen  10  more,  and 
killed  in  the  same  proportion,  he  would  have  brought 
home  6  :  how  many  squirrels  did  he  see? 

72.  Two  men  hired  a  pasture  for  $33,  and  agreed 
that  the  pasture  of  2  cows  should  count  for  1  horse: 
one  pastured  4  cows  and  2  horses  for  3  weeks,  the 
other  2  cows  and  3  horses  for  2  weeks :  how  much 
should  each  pay  ? 

73.  If  40  dollars  be  divided  between  two  persons, 
so  that  one  shall  have  3  dollars  every  time  that  the 
other  has  2,  how  much  will  each  receive  ? 

74.  In  a   school  of  44  pupils,  there  are  If  times 
as  many  girls  as  boys:  how  many  of  each? 

75.  A  man   bought  a  horse,    saddle,  and  bridle : 
he  paid  2  times  as  much  for  the  saddle  as  for  the 
bridle,  and   11    times  as  much  for  the  horse  as  for 
saddle  and  bridle  both;  in  all  he  paid  $108:  how 
much  did  he  pay  for  each  ? 

76.  A  merchant  sold  6|  yards  of  cloth  at  4  dollars 
a  \  ai-d,  and  took  his  pay  in  equal  quantities  of  rye 
und  wheat,  the  former  at  50  cents,  and  the  latter  at 
$1  a  bushel :  how  much  wheat  did  he  receive  ? 

77.  Find  the  ages  of  three  persons,  knowing  that 
the  age  of  the  second  is  equal  to  twice  the  age  of  the 
first,  and  the  age  of  the  third  five  times  the  age  of 


LES.  XVIII.]          INTELLECTUAL    ARITHMETIC.  171 

the  first  and  second,  and  that  the  sum  of  their  ages 
is  90  years. 

78.  A  drover  after  selling  J  of  his  flock  of  sheep, 
finds  that  if  he  had  sold  4  less  he  would  have  sold 
just  \  of  his  flock  :  how  many  had  he? 

79.  A  boy  being  asked  his  age,  replied  :  if  to  my 
age  $ou  add  \  of  it,  \  of  it,  and  14  years,  you  will 
have  a  sum  equal  to  3  times  my  age. 

80.  A  hare  pursued  by  a  hound,  runs  13  times  as 
far  as  the  distance  between  them  when  the  pursuit 
commenced,  and  the  hound  runs  28  rods  before  over 
taking  the  hare  :  how  far  was  the  hare  in  advance 
when  the  pursuit  began  1 

81.  A  farmer  buys  a  pig,  for  which  he  pays  3  dol 
lars,  and  also  a  sheep  and  a  cow  ;  the  cow  cost  3 
times  as  much  as  the  pig  and  sheep,  and   the  sheep 
cost  5  times  as  much  as  the  pig  :  what  was  the  cost 
of  each  1 

82.  A  garrison  of  300  men,  has  provisions  for  6 
months,  at  the  rate  of  16  ounces   a  day  :  how  much 
must  the  allowance  be  diminished  to  last  8  months'? 

83.  If  a  staff  3  feet  high,  casts  a  shadow  6  feet  in 
length:  how  long  is  a  pole  which  casts  a  shadow  20 
feet  long,  at  the  same  time  of  day? 

84.  If  a  stick  12  feet  long  casts  a  shadow  2  feet  long, 
what  is  the  length  of  a  pole  which  casts  a  shadow  9 
feet  long,  at  the  same  time  of  day  ? 

85.  A  man's  coat  cost  him  1|  times  as  much  as  his 
pantaloons,  his  pantaloons  cost  6  dollars  less  than  his 
coat:  what  was  the  cost  of  the  pantaloons]     What 
was  the  cost  of  the  coat  ] 

86.  James,  John,  and  Charles  together  have  150 
marbles;  Charles  has  ^   as  many  as  John;  if  John 
gives  Charles  30  of  his,  they  will  all  have  the  sanio 
uumber  :  how  many  has  each  1 

87.  A  cistern  has  two  cocks,  one  of  which  will  fill 


i72  INTELLECTUAL    ARITHMETIC.          [SEC.     17 II. 

it  in  2i  hours,  and  the  other  in  3^  :  in  what  time  will 
they  fill  it  running  together  ? 

88.  A  person  hired  a  man  and  two  boys.     To  the 
man  he  gave  six  shillings  a  day,  to  one  boy  four,  and 
to  the  other  three ;   at   the  end   of  the  time  he  paid 
them  104  shillings:  how  long  did  they  work? 

89.  A  cask  of  wine  leaked  out  one  quarter,  after 
which  one  third  of  the  remainder  was  drawn,  when 
the  cask  vvas  found  to  contain  80  gallons  :  how  much 
did  the  cask  hold  ? 

90.  A  market  woman  bought  a  certain  number  of 
fggs  at  3  for  2  cents,  and  an  equal  number  at  5  for 
4  cents.     She  paid  for  both  lots  44  cents:  how  much 
did  her  eggs  cost  her  apiece,  and  how  many  did  she 
buy  1 

91.  A  market  woman  bought  a  certain  number  of 
eggs  at  the  rate  of  4  for  3  cents,  and  sold  them  at  the 
rate  of  5  for  4   cents,  by  which  she  made  4  cents. 
What  did  she  pay  apiece  for   the  eggs?     What  did 
she  make  on  each  egg  sold  ?     How  many  did  she  sell 
K>  make  4  cents? 

92.  A   market  woman  bought  36  fowls,  of  three 
different  sorts,  for  which  she  paid  84  shillings.     She 
bought  half  as  many  of  the  first  sort  as  of  the  second, 
and  three  times  as  many  of  the  third  sort  as  of  the 
first,  and  paid  1,  2,  and  3  shillings  apiece  for  each 
sort :  how  many  of  each  did  she  buy  ? 

93.  If  James  can  weed  his  father's  onions  in  9 
hours,  and  John  in  12  hours,  how  long  will  it  take 
both,  working  together,  to  weed  them  ? 

94.  A  lady  wishes   a  dress,   and  does  not   know 
whether  to  buy  silk  or  muslin.     The  silk  costs  9  shil 
lings  a  yard,  and  the  muslin  3.     If  she  purchases  the 
silk  it  will  cost  72  shillings  more  than  the  rnusliii: 
how  many  yards  did  she  need  ? 

95.  A  boy  was  hired  for  30  days.     For  every  day 


LE8    rVIII.]        INTELLECTUAL    ARITHMETIC.  173 

he  worked  he  was  to  receive  5  dimes,  and  for  every 
day  he  played  he  was  to  pay  2  dimes  for  his  board. 
When  he  came  to  settle  he  received  87  dimes  :  how 
many  days  did  he  work  ? 

96.  How  many  small  cubes,  1  inch  on  a  side,  can 
be  saviad  out  of  a  cubic  block,  2  feet  on  a  side,  allow- 
ing  no  waste  in  sawing1? 

97.  If  A  can  do  J  of  a  piece  of  wor*k  in  2  days,  and 
B  can  do  J  of  it  in  4  days, 'how  much- of  it  can  each 
do  in  1  day,  and  h6w  long  will  it  take  both  to  do  it, 
working  together  1 

98.  A  man   having  a  goose,  pjg,   and  calf,    was 
asked  the  value  of  them.     He  said  that  the  three  were 
worth  30  shillings,  that  the  goose  was  worth  one  third 
/is  much  as  the  pig,  and  that  the  calf  was  worth  1-J- 
times  as  much  d£  the  goose  and  pig  together  :  what 
was  the  value  of  each  ? 

99.  James  is  10  of  his  own  pices  behind  John,  arid 
in  pursuit  of  him.     James  steps  3  times  while  John 
steps  4  times ;  but  James'  steps  are  twice  as  long  as 
John's  :  how  many  steps  must  James  make  to  pver 
take  him  ?  ^  <v 

100.  Two  families  bought  a  barrel  of  flour  together/ 
for  which  they  paid  $8,  and  agreed  that  each  child 
should   count  half  as  much  as  a  grown  person.     In 
one  family,  there  were  3  grown  perspijg  and  3  children ; 
and  in  the  other,  4  grown  persons  and  10  children ; 
the  first  family  fed  from  the  flour  2  weeks,  and  the 
aecoud  3  :  how  much  ought  each  to  pay  1 

15* 


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